Joe Joyce wrote earlier:
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"I like this design [Hyperchess4] because it is humanly playable, looks like chess, and gives people the feel of 4D and the pieces the freedom to move through 4D space without overwhelming the players with 3D and 4D diagonals, or very many of the available 2D diagonals, for that matter. Most who design 4D chess variants using the 2D layout of 2D boards to represent 4D space give the pieces moves that are based on a 4D space that is 2D x 2D = 4D. I've found this creates a totally chaotic game, where the state of the board cannot be reasonably projected even 2 or 3 turns into the future. Pieces move so freely that in a couple moves, they can be anywhere on the board, generally by many paths which can't all possibly be guarded in 2 or 3 moves. So I get my 4D by basing movement on a 2D + 2D = 4D concept. It gives you the same 4D game space, but it restricts movements to fairly easily visualizable, fairly simple patterns that players can project 2 - 3 moves into the future. While it is fast-moving, it is not chaotic."
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Interesting. By my estimate, in a hypothetical worst case position, one side in a game Hyperchess4 might have around 200 legal moves available, if I've got the rules correctly. My earliest effort (4*Chess) to produce a 4D variant is based on a 4D space that is indeed 2D x 2D = 4D, and by contrast it might have around 600 legal moves available in such a worst case, so that a computer (or possibly a human player) could be able to look about 4 ply ahead in Hyperchess4 for every 3 ply looked ahead in playing my variant, I estimate. However, that's without taking into account the possible human difficulty with visualizing moves in 4D chess that you've noted.