A color bound bishop in 3 dimensional chess can travel in 12 different directions, which ostensibly makes it more powerful than the rook, which can travel orthogonally in only 6 possible directions. There then remains the 8 space diagonal directions. Possibly assign them to the rook to increase its power beyond the bishop to keep the relative strengths in line with 2D chess. We can't assign the space diagonals to the bishop or else the piece will no longer be color bound.
In higher dimensions more directions are possible. I think it remains possible to assign color bound directions to the bishop and color changing directions to the rook.
In 2 dimensions, the knight can jump to the 5^2-2*3^2+1 = 8 squares in a 5x5 neighborhood that a queen cannot reach. Extending to 3 dimensions, the knight can jump to 5^3-2*3^3+1 = 72 different cubes, making it a very powerful piece. In higher dimensions, the knight becomes relatively more powerful.
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