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What do you do about this? In another guise, this problem of chaos is
addressed as the slippery king problem. How can you interdict the king in
higher dimensions? In 2D, the king has 8 moves, in 3D, it's 26, and in 4D,
81, assuming the king moves 1 square 'in any direction'. And, because
checkmate is a specific rule for a specific piece, it poses a different
problem than the general problem of chaos, which is being unable to
forecast the board, and thus, unable to choose the best moves. So 2 related
but different problems must be solved for a humanly-playable game.

A number of approaches have been taken. The most obvious is to restrict the
number of dimensions in the game. I do not recall seeing any chess variant
described of higher than 6 dimensions. Most games are 3D, but there are a
growing number of 4D games, including Alice Raumschach. 'Alice'-ing, or
Alicing, is itself a nice cheat, a mutator that adds a dimension to any
boardgame, without all the associated problems of the extra diagonals. They
are all subsumed in seeing if a square is available to land on, in that
higher dimension [along with a little associated confusion about en
passant, as well]. The game is played as the lower-dimensional game,
effectively. But this approach is too limiting. Players are looking for
that extra dimension of freedom of movement.