Here are some pieces that may be substituted for the Bishop.
The Crooked Bishop has its own chapter.
Mathematically, this piece should be noticeably stronger than the Bishop, but it has a certain built-in awkwardness that has earned it the name of d***ed DAmf. In practice, it is probably slightly weaker than the Bishop.
This piece has been tested in a game, which is not nicely-formatted yet. The DAmF was found to be acceptable in strength, but unintuitive to use; that is, chessplayers find it easy to use the Augmented Knights, but difficult to use the DAmF.
Although the result of a single game is not enough to prove the strength of a piece, it seems very likely that the DAmf can be used as a substitute for the Bishop, and that neither side will have a big advantage from this confrontation.
Mathematically, this piece should be clearly stronger than the Bishop, but in its one test between two masters, the Bishops did rather well. Although the result of a single game is not enough, it seems that the DAF is in general slightly stronger than the Bishop, but not enough to decide the game or create a tangible advantage. However, if you can keep both DAFs on the board while playing an endgame of attrition, sooner or later they will probably run amok and produce a sudden win by checkmate; this is not a bad advantage to have.
This is actually a very important finding. (January 1996) It means that the existing means of measuring the values of chess pieces are already more precise than they really need to be, even when both players in the game are masters. Thanks to this, you can expect to see many more entries here soon.
If one side had the DAF and the other side had the DAmF, would the difference in value be enough to win the game, or at least to produce a tangible advantage?
The cRmF moves one square diagonally, and captures like a Rook; it may not move like a Rook, and it may not capture diagonally.
This piece seems good, but is not approved for tournament play. :-)