If you take the slice parallel to the cubic lattice coordinate planes, you get a normal chess board, and 3d bishops project to normal Bishops on it. As do Unicorns.
The way I'm thinking about it, when you project along one of the 3d orthogonal directions, you do get a normal chess board, but 3d bishops become 2d queens, and unicorns become 2d bishops. E.g., projecting along the z direction, a 3d bishop sliding along (+1,0,+1) projects to a rook movement (+1,0).
The way I'm thinking about it, when you project along one of the 3d orthogonal directions, you do get a normal chess board, but 3d bishops become 2d queens, and unicorns become 2d bishops. E.g., projecting along the z direction, a 3d bishop sliding along (+1,0,+1) projects to a rook movement (+1,0).