Unless I'm mistaken, the various 5 levels of these 3D Hexagonal variants should have their hexes alternating different colour patterns at different levels. That's so that bishop moves between levels would stay on the same coloured hexes as the hex that the bishop would start on, and a rook would change the hex colour it's on if it's moving up or down one level (granted, it may take a significant amount of extra effort to show alternating colour patterns like that).

Note that a 5 level 3D Hexagonal variant that's a fairly close analogue of Raumschach (classic 3D chess variant that uses square cells) would be possible (such a variant might be called 'Hexagonal Raumschach'). Such could be with 12 pieces (K,Q,2Rs,2Ns, 3Bs and 3Us) and 15 pawns for each of the 2 players, having 5 small (with a peak width of 7 hexes, and 37 hexes) 2D hex-shaped board levels, with the two side's armies setup on 3 levels each, and Glinski's pawn rules (except no double-step) in effect for the possible pawns movements that are just on a single 2D board. Pawn moves between levels would be single step rook or forward direction bishop (or unicorn, to be truer to Glinski[?]) moves, the latter in case of capturing. That's with unicorns being seemingly superior pieces than in the case of Raumschach, in a way, since I believe just three of them on different colours would be able to eventually reach every cell on a such a 3D Hexagonal board, if empty. Also, a rook could move to a greater number of cells (than in the case of Raumschach) on an empty 3D Hexagonal board that has at least as many cells as a Raumschach board (yet having no more than 5 levels).

The chief (and major) drawbacks of such 3D Hexagonal variants (compared to Raumschach) might be that a number of piece types' moves between levels would be more difficult to visualize or concisely describe in detail for the players, perhaps more than is the case even for 4D variants that have square cells. Such 4D variants also have the positive features that a 4D movement piece type is possible besides the (3D) unicorn, i.e. the (4D) balloon, and that a single unicorn can eventually reach every cell on such an (empty) 4D board. [edit: I'd tentatively estimate the piece values for a variant 'Hexagonal Raumschach', with the general specifications for it as described above, to be: P=1; U=2; B=3; R=3; N=4; Q=10, and a K would have a fighting value of 4.5 (noting it can't be traded). Below is a diagram of one possible setup for such a variant; adding additional diagrams to illustrate piece movements for players, in a [preset] submission, seems a lot of extra work for something that may not be entirely viable {P.S.: I've since decided to make a submission based on the diagram}:]

I've also been thinking about possible 4D Hexagonal variants; perhaps it's a matter of taste, but for such 4D variants there seems no really positive points of comparison to 4D variants that use square cells (note again that, in either case, a single unicorn is able to reach every cell on an empty board eventually). For one thing, I think a minimum of 9 Balloons (these are exclusively 4D-moving pieces), instead of 8, would be needed to reach every cell on a 4D Hexagonal board. That's with, say, 37+ 2D boards arranged like a hex; each might be hex shaped with peak width 7 hexes, and 37 cells, easily allowing e.g. 2 armies of 36 pieces and 45 pawns each in the case of 37 2D boards. A more manageable configuration (in terms of diagramming, at least) would be 19 2D boards arranged like a hex, each hex shaped with peak width 5 hexes, and 19 cells, easily allowing e.g. 2 armies each of 36 pieces and 20 pawns (all within 4 or 5 moves of promoting on a last row 2D board). All of this is in the case of a 4D Raumschach analogue.

[edit: It appears to me now that pieces with a Balloon movement power would make infeasible variants based on such 4D Hexagonal games that include them, anyway, if also having multple pieces on the 2D boards in the players' camps. That's as such pieces' scope may way too soon reach too deep into the opposition's camp, on some 2D board(s). Any sort of Hexagonal analogue of TessChess (i.e. being 4D) seems infeasible, or at least unattractive, if only due to the possibility of a fast knight or bishop/queen balloon style check (or attack on an enemy queen) - or due to the use of an ugly and/or very large setup designed to try to avoid or minimize the effect of such. A Hexagonal analogue of Hyperchess4 (i.e. being 4D) may be feasible, however. A Hexagonal analogue of 4D Quasi-Alice Chess also might be viable, noting, however, that the pawns in front of each rook would be prone to immediate assault and loss if a Glinski's or McCooey's Hexagonal Chess setup for an 11x11 (2D) Hexagonal board is used (in addition, bear in mind the paragraph below). P.S.: I later submitted to CVP a 4D Hexagonal variant based on Hyperchess4.]

Similarly, it seems to me there aren't really any positive points of comparison between Alice Chess (classic 3D variant that uses square cells, on two 2D boards) and possible Hexagonal versions of it. That's except that a possible major plus is that knights are not colour-bound on either 2D Hexagonal board (though then knights as well as bishops ought to be worth more than 4 pawns, meaning that they seldom if ever can be equitably exchanged for a small number of pawns, which might be seen as a major minus; this applies to my own 4D Quasi-Alice Chess variant, too). A couple of such possible variants might be appropriately dubbed 'Glinski's Hexagonal Alice Chess' and 'McCooey's Hexagonal Alice Chess'. Note that in further comparison, standard Alice Chess has some possible early mating traps, which seems quite desirable, not to mention that all the basic mates available in standard chess are possible in Alice Chess (although, these things could be said when comparing any number of 2D Hexagonal variants to standard chess, but note for Alice Chess it's possible there might be significantly more early mating traps than for chess, which may well not similarly apply to an Alice Chess-style version of a given Hexagonal variant).

Also related to possible Hexagonal variants, I recently thought of using a ring board (as in the square board variant Rococo that's popular on CVP/Game Courier) on a 91 cell hex-shaped board with hex cells, to try to invent some sort of fresh variant. The concept seems worthless, however, as an attempted 2 hex diagonal jump by a piece over a diagonally adjacent opposing piece on the edge of the inner cells can sometimes go past the ring of outer cells. Nevertheless, such total failures are reassuring in a way, in that they show that there can be limits that otherwise plausible variant ideas may exceed (otherwise all of the many variant ideas I've come up with to date might be viable to some extent).

Raumschach

Hexagonal Raumschach

Glinski's Hexagonal Chess

McCooey's Hexagonal Chess

Alice Chess

TessChess

Hyperchess4

4D Hexagonal Chess

4D Quasi-Alice Chess

Rococo