However, I would say that real chess is sort of "2.5" D by this definition rather than full 3D, since the turn number coordinate of every piece can only go up, not down.
[1]: The game's formalism is slightly complicated by the fact that a piece could make a move of (x, y, t, L) -> (x, y, t - 1, L) which would count as moving one square in one dimension, but then the game's "system" automatically transforms that move into a (x, y, t - 1, L +/- N) move. (The choice of +/- depends on which player made the move. The value of N depends on how many other timelines there are at that t.) This happens because of the rule that the past cannot be changed, so altering the past requires branching the timeline.
Similar things also happen with a move of (x, y, t, L) -> (x, y, t, L2), because within a single turn there is a concept of whether the board has already had a move played on it or not. So if the board at L2 has already had a move played on it, this new move is instead transformed into (x, y, t, L3). So it counts as a move of (L2 - T) squares for the purpose of validity, but is actually a move of (L3 - T) squares.