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Daniil, you are right about the difficulties of visualization. There are
two inherent problems of higher-dimensional chess. One is the slippery
king; without significant restrictions, the king is exceedingly difficult
to mate. The other is the sheer number of directions that a piece can move.
On a 2D board, the Q moves in 8 directions; on a 3d, 26 directions, and on
a 4D board, a 4D queen can move in 80 different directions, and a standard
knight move hits up to 48 cells on a large enough 4D board. How do you
project several moves into the future? How do you even predict what your
opponent will do next turn? If you limit the moves of all the pieces in
higher D games, this can make both problems much more tractable. For 4D,
Hyperchess and TessChess provide a nice pair of examples, as they are very
well matched.

Charles, one observation from playing on the Hyperchess board - the
same-color patterning of the 2D 'levels' helps in both 'seeing' the
moves and in putting the piece down on the correct location. It's a lot
easier to hit the wrong square on a 2D layout of a '4D' board than you
might think, odd as that sounds.