'Really diagonal is just orthogonal on a different, bigger board'
This is something that I have illustrated with my Nested series of variants. For the implication in 3d, see my comments on Tetrahedral Chess.
'Knights are diagonal but use 2 different diagonals together that make them not colorbound'
Not technically diagonal but I see what you mean. The moves of the Veering Knight and Backing Knight are again the orthogonals of a smaller board:
.*....*.. ..*....*.
...*....* *....*...
*....*... ...*....*
..*....*. .*....*..
....@.... ....@....
.*....*.. ..*....*.
...*....* *....*...
*....*... ...*....*
..*....*. .*....*..
'hunters (pieces that move and capture in diferent ways)'
It is snipers that have different noncapturing and capturing moves; hunters have different forward and backward moves (and no same-rank ones).