The D&D Expert rulebook contains a Part 9: Special Adventures, covering waterborne adventures, including encounters and combat at sea.
Combat on the Astral Plane, from Deities & Demigods (1980)
I would include a similar section in a Companion supplement, covering underwater and extraplanar adventures.
This might involve how certain spells are affected underwater, or on other planes (perhaps using the table in Q1: Queen of the Demonweb Pits).
"Cloud World" from Q1 "Queen of the Demonweb Pits" (1980). Illustration by Erol Otus.
Unsuspected by all but the most powerful magic-users (M26+), beyond the multiverse are alternative realities, called "dimensions." It is guessed that other dimensions coexist in the same space as the multiverse, yet cannot be detected by mortal perceptions. Even Immortals, with their heightened perceptions, may have limited powers in alternative dimensions. One such dimension is the Nightmare Dimension, which is inhabited by creatures that haunt the dreams of mortal and Immortal alike.
Past all planes and dimensions lies a mysterious barrier, called the Dimensional Vortex, home to horrific monsters called "vortex creatures." Here, even the Immortals fear to tread.
A D&D Companion supplement should include information on other planes, involving concepts which evolved in the transition from OD&D to AD&D, although diverged in the BECMI line with publication of the D&D Master and Immortals Sets.
The Inner Planes, from the AD&D 1e Deities & Demigods Cyclopedia (1980)
The Ethereal and Astral Planes were introduced in the Eldritch Wizardry supplement, and further defined in "Planes: The Concepts of Spatial, Temporal, and Physical Relationships in D&D" by Gary Gygax, published in The Dragon #8 (July, 1977)*
*see OD&D Cosmology: The Inner Planes of Existence; Notes on the Ethereal Plane; OD&D Cosmology: The Outer Planes; The Astral Plane
AD&D cosmology was described in Appendix 1: The Known Planes of Existence, in Deities & Demigods (the Inner Planes were covered in more detail in articles by Len Lafolka in Dragon #42 (October, 1980) and by Gary Gygax in Dragon #73 (May, 1983)
Information on the Inner Planes is essential, including the Positive and Negative Material Planes (not covered in the D&D Companion Set) as well as the paraelemental planes. Wandering Monster tables for the various planes should also be included.
The Outer Planes were to be covered in the D&D Master Set (1985) which focused instead on introducing a cosmology involving five Spheres of Power. (I would exclude the AD&D outer planes, since they are linked to the AD&D alignment system.)*
*alternately, certain AD&D outer planes could be included as pantheon-based, as opposed to alignment-based outer planes
The Astral and Outer Planes were covered in more detail in the Immortals Set (1986), although the nature of the Astral Plane was revised. I would favour a more OD&D/AD&D based conception of the Astral Plane, and its relationship to dimensions.
The "Domain Game" has it's roots in Dave Arneson's original Blackmoor campaign, and was expanded upon in the D&D Companion Set.
Illustration by Erol Otus, from the World of Greyhawk folio (1980).
The Cook/Marsh Expert rulebook briefly covers taxes (10 gp per year or whatever the DM decides). The Companion Set provides rules for populations and resources.
Additional rules involving administration (income, experience points, expenses, troops), tournaments, confidence level, and events* are also included.
*see also Arneson's "Chance Cards"
I think it's important to have rules for domain management, even though most of the time, these will merely serve as a backdrop to the main action in a campaign.
The Master Set rules include a lengthy section on artifacts, describing four categories (minor, lesser, major, greater) along with general characteristics, powers, and adverse effects.
Table from D&D Master Set (1985)
Several examples drawn from real world cultures and mythology were described.
The Rules Cyclopedia omitted the tables for determining powers and adverse effects, and included a single, new example (the Mask of Bachraeus)
I would include the Master Set rules for creating artifacts in a Companion supplement, along with appropriate tables, but use a different example.
One possibility are the Crowns, Orbs, and Sceptres of Might:
There are several sets of the Crown, Orb and Scepter, one for each alignment. They are scattered across the world and well hidden. Each item is only useful to one who is of the same alignment as the item, others take 2-24 points of damage when touching it.
from Eldritch Wizardry (1976)
These three complete sets bestow great powers, but even mere possession of a Crown of Might gives a character of the same alignment great benefits (if a character of another alignment touches such an item he or she takes 5-30 points of damage and must save vs. Spells or be instantly killed).
Note: Each of these items is so similar in appearance to the other that examination will reveal no difference, and magical detection will not reveal their alignment.
Crowns of Might
Each Crown is a slender diadem of gold set with three precious stones of great size so as to bring 50,000 or more gold pieces if openly sold. While being worn, the Crown raises its wearer's level of experience by one and confers additional powers/effects.
Should a character wearing a Crown touch an Orb of Might or a Sceptre of Might of a different alignment, he or she takes damage and must save as noted above, and if the saving throw is successful, one penalty will affect him or her. However, if the Orb or Sceptre is of the same alignment, extra powers (and effects) are conferred upon the possessor.
Orbs of Might
Each Orb is of platinum, encrusted with gems, and topped with a device of precious metals and stones, so as to be worth 100,000 or more gold pieces on the open market. Each Orb is equal to a Gem of Brightness and has additional powers.
If a character of another alignment touches an Orb different from his or hers, a save vs. Spells must be made to avoid death and from 4-24 points of damage will be taken if the save is successful. If the character so touching an Orb also possesses a Crown and/or Sceptre, surviving the save vs. Spells will invoke a penalty.
For additional powers in combination with other items of regalia, see Crown of Might.
Sceptres of Might
Each Sceptre is wrought of bronze inlaid with silver and many fine gems, with a huge precious stone tipping its 2' length, giving the item a value of 150,000 or more gold pieces on the open market. Each Sceptre functions a Rod of Beguiling and has additional powers.
The effects of handling a Sceptre of an alignment not the character's own are the same as those of a Crown.
For additional powers in combination with a Crown or Orb of the same alignment, see Crown of Might.
"Part 7: Treasure" in a Companion supplement should include expanded tables for each of the categories described in B/X, like those presented in the D&D Companion Set rules.
Illustration by Bill Willingham, from the Monster & Treasure Assortment, Sets One-Three: Levels One-Nine (1980)
Expanded tables for magic items were also included, covering swords, armor, miscellaneous weapons, potions, scrolls, rings, wands/staves/rods, and miscellaneous magic.*
*I would probably expand the miscellaneous magic table, in order to include a greater variety of items originally described in OD&D or AD&D
Aerial Servant - from AD&D 1e Monster Manual/Companion Set
Beholder - from Greyhawk/AD&D 1e Monster Manual/Companion Set
Catoblepas - from The Strategic Review #7/AD&D 1e Monster Manual/"Nekrozon" in Master Set
Illustration from Deities & Demigods (1980)
Couatl - from Eldritch Wizardry/AD&D 1e Monster Manual
Demon - from Eldritch Wizardry/AD&D 1e Monster Manual/Master Set
Demon Rulers (Orcus, Demogorgon) - from Eldritch Wizardry/AD&D 1e Monster Manual/Master Set (include details on the Wand of Orcus, with reference to the Artifacts section)
Djinni (Greater)/"Pasha" in Companion Set
Dolphin - from Blackmoor/AD&D 1e Monster Manual/Companion Set
Dragon (Crystal, Onyx, Jade, Sapphire, Ruby, Brown) - from Companion Set
Dragon Rulers (Pearl, Opal, Diamond; The Great Dragon) - from Companion Set
Eel - Electric, from AD&D 1e Monster Manual/X7; Giant, from Blackmoor/AD&D 1e Monster Manual/X7; Weed, from Blackmoor/AD&D 1e Monster Manual/AC9 Creature Catalog
Efreet (Greater)/"Amir" in Companion Set
Elf, Aquatic - from Blackmoor/AD&D 1e Monster Manual/AC9 Creature Catalog
Eye, Floating - from Blackmoor/AD&D 1e Monster Manual
Frog - Giant, from Blackmoor/AD&D1e Monster Manual; Killer, from AD&D 1e Monster Manual/DA2; Poisonous, from AD&D 1e Monster Manual/AC9 Creature Catalog
Gas Spore from AD&D 1e Monster Manual/"Blast Spore" in Companion Set
Ghost - from The Strategic Review #3/AD&D 1e Monster Manual/Companion Set
Golem (Rock, Silver) - from AC9 Creature Catalog
Hag - Black, from Master Set; Sea, from Blackmoor/Master Set
Hippocampus - from AD&D 1e Monster Manual
Homonculous - from the Greyhawk errata/AD&D 1e Monster Manual
Illustration by David S. LaForce
Ixitxachitl - from Blackmoor/AD&D 1e Monster Manual/"Devilfish" in Companion Set
Jellyfish, Giant (Marauder, Man-O-War) - from Blackmoor/X7
Ki-rin - from Eldritch Wizardry/AD&D 1e Monster Manual
Lamia - from AD&D 1e MM/"Lamara" in AC9 Creature Catalog
Lammasu - from Greyhawk/AD&D 1e Monster Manual
Lich - from Greyhawk/AD&D 1e Monster Manual/Master Set
Locathah - from Blackmoor/AD&D 1e Monster Manual/"Kna" in AC9 Creature Catalog
Manta Ray (Normal, Giant) - from Blackmoor/X7/Companion Set
Masher - from Blackmoor/AD&D 1e Monster Manual/AC9 Creature Catalog
Mind Flayer - from The Strategic Review #1/AD&D 1e Monster Manual
Morkoth - from Blackmoor/AD&D 1e Monster Manual/"Mesmer" in AC9 Creature Catalog
Naga (Water, Spirit, Guardian) - from The Strategic Review #3/AD&D 1e Monster Manual
Nightmare - from AD&D 1e Monster Manual/AC9 Creature Catalog
Ogre Mage ("Oni") - from Greyhawk/AD&D 1e Monster Manual
Phase Spider - from Greyhawk/AD&D 1e Monster Manual/"Spider, Planar" in Master Set
Phoenix - from Master Set
Rakshasa - from The Strategic Review #5/AD&D 1e Monster Manual/"Randara" in AC9 Creature Catalog
Illustration by Erol Otus, from the AD&D 1e Rogues Gallery (1980)
Roper - from The Strategic Review #2/AD&D 1e Monster Manual/XL-1
Sahuagin - from Blackmoor/AD&D 1e Monster Manual/"Shark-Kin" in AC9 Creature Catalog
Sea Horse - from Blackmoor/AD&D 1e Monster Manual/X7
Sea Serpent (Greater) - from X7
Shambling Mound - from The Strategic Review #3/AD&D 1e Monster Manual
Shedu - from Eldritch Wizardry/AD&D 1e Monster Manual
Slithering Tracker - from The Strategic Review #5/AD&D 1e Monster Manual
Slug, Giant - from Greyhawk/AD&D 1e Monster Manual/Master Set
Sphinx - from AD&D 1e MM/Master Set
Strangle Weed - from Blackmoor/AD&D 1e Monster Manual/AC9 Creature Catalog
Su Monster - from Eldritch Wizardry/AD&D 1e Monster Manual
Titan - from Greyhawk/AD&D 1e Monster Manual
Triton - from Greyhawk/AD&D 1e Monster Manual/X7
Umber Hulk - from Greyhawk/AD&D 1e Monster Manual/"Hulker" in AC9 Creature Catalog
Will O' Wisp - from Greyhawk/AD&D 1e Monster Manual/"Wychglow" in AC9 Creature Catalog
Wind Walker - from The Strategic Review #3/AD&D 1e Monster Manual
Xorn - from AD&D 1e Monster Manual
The monster section in the Moldvay Basic rulebook has 70 entries, while the Cook/Marsh Expert rulebook has 67, so a Companion supplement should include a comparable number.
The AD&D 1e Monster Manual introduced the concept of magic resistance, similar to spell immunity as described in the Companion Set rules/anti-magic effects in the Master Set rules.
The Master Set rules also included guidelines for changing monsters, intelligence of creatures, spell casters (non-human), and undead lieges and pawns.
Illustration by Erol Otus, from the AD&D 1e Fiend Folio
OD&D:
All of the monsters in Monsters & Treasure are listed in B/X.
Greyhawk:
Monsters not included in B/X include druids, tritons, ogre magi, titans, will o'wisps, liches, the metallic dragons (brass, copper, bronze, silver), the platinum "king" and chromatic "queen" dragons, lammasu, beholders, umber hulks, phase spiders, giant tics, giant slugs, homunculi, and golems (flesh, stone, iron).
I would include most of these in a Companion supplement, except the druid (available in other sources), the metallic dragons (the brass dragon was included in Holmes), platinum "king" and chromatic "queen" dragon rulers (those in BECMI are different), giant tic (included in Holmes), and golems (B/X has its own versions).
Blackmoor:
Monsters not included in B/X include giant frogs, giant beavers, giant otters, giant wasps, fire lizards, minotaur ("monitor"?) lizards, elasmosaurus, mososaurus, giant eels, lamphrey, sea horse, and the Portugese man-of-war.
Those attributed to Steve Marsh include dolphins, aquatic elves, pungi ray, manta ray, water spiders, weed eels, sahuagin, floating eyes, ixitxachitl, locathah, morkoth, and mashers.
Eldritch Wizardry:
I would include demons, which appear in the Immortals rules. I would also include couatl, ki-rin, shedu, and su-monsters, as well as mind flayers (from The Strategic Review #1), but not brain moles, cerebral parasites, intellect devourers, or thought eaters (created to counter psionics).
AD&D 1e Monster Manual:
The AD&D 1e Monster Manual collected all of the creatures in OD&D + supplements, most appearing in The Strategic Review + early editions of The Dragon, and included some new monsters, as well.
Among these, possible candidates for a Companion supplement include the aerial servant, catoblepas (from The Strategic Review #7), gas spore, hippocampus, lamia, naga (from The Strategic Review #3), nightmare, rakshasa (from The Strategic Review #5), roper (from The Strategic Review #2), shambling mound (from The Strategic Review #3), slithering tracker (from The Strategic Review #5), sphinx, wind walker (from The Strategic Review #3), and xorn.
D&D Companion Set:
The Companion Set rules included some original monsters (mostly those from other planes), but also many creatures who appeared previously in the new monster section of various modules.
D&D Master Set:
The D&D Master Set rules included many original monsters, in addition to some reskinned from OD&D (ie. the ixitxachitl as "devilfish"; and catoblepas as "nekrozon").