The Bishop is a color-bound and long-range piece of a certain value. A Bishop substitute must have the correct value, of course; but in addition, in order for it to feel like a Bishop, it should be either a long-range or a color-bound piece (or both!).
Unfortunately, the basic geometrical pieces, to which I have thus far devoted most of my efforts, are not long range. Recently, I have started looking at strange long-range pieces, and have invented a piece which I think may be completely new:
In fact, from e4 there are 8 possible paths it could take: e4-d5-e6-d7-d8, e4-f5-e6-f7-e8, e4-f5-g4-h5, e4-f3-g4-h3, e4-f3-e2-f1, e4-d3-e2-d1, e4-d3-c4-b3-a4, and e4-d5-c4-b5-a4.
Note: It is not legal for it to go e4-d5-e6-f7-e8, because the movement from e6 to f7 would fail to make a 90 degree turn.
Note: Remember that it is exactly like a Bishop except that it must turn; it cannot go through occupied squares.
This piece is surprisingly weak, at least in the opening and middlegame and early endgame; either that, or I don't know how to use it well yet! Quite possibly, it becomes very strong in the late endgame; a pair of them in a really open position would be deadly indeed.
The first reason for this seems to be that, although it moves in 8 different directions, there are only 4 different squares for its first step, and 4 different squares for its second step. Only when it moves three squares or further does it attack 8 different squares, and even then it is only for odd-numbered distances that it attacks 8 squares. Because the standard chessboard is only 8 squares long in any direction, the only time that the Crooked Bishop can attack 8 different squares at the same distance is when it is on one of the 4 center squares. The point of all this is that the number of directions a piece moves is extremely important to its value, and because the 8 different crooked paths share so many squares, the Crooked Bishop does not really gain the benefit of moving in 8 different directions.
The second reason for this is that the normal positions that we know how to get do not offer the Crooked Bishop much scope. As chessplayers, we are trained to create open files and diagonals, but we have no reason to learn about zigzags.
If Black has Crooked Bishops, White has normal Bishops, and Black is slightly experienced with this piece, Black's advantage should be not much larger than the advantage given to White in the usual game of Chess.
I have looked at the game where Black has Crooked Bishops on f8 and c8, and Waffles on g8 and b8, while White has the standard Chess army. It seemed to work fairly well at first, but now I wonder about it; the more I play this game, the more danger I find for White.
Even so, I have not gotten to be so skilled with the Crooked Bishop that this game is a near-win for Black; only a tangible advantage. What I fear is that increasing skill would tilt the balance even more.
Especially interesting would be a piece with the choice of moving either as a Crooked Bishop or as a normal Rook; with so much of its power duplicated (from d1, it has 3 different ways to reach d5, but all 3 paths must cross d3), it must be noticeably weaker than the normal Queen.
A piece that moves as Crooked Bishop or normal Bishop would be interesting, much stronger than a Rook but also much weaker than a Queen; and presumably weaker than the Bishop+Knight combination.
This piece must be very strong, but it is extremely hard to get it developed. For example, suppose White has one on a1 in the opening position: 1. b4 e5 2. a4 B:b4 3. zRa1:e5 gets it out of the corner, but at a great positional cost.
In the opening position, a Crooked KnightRider on b1 can play 1. zNNb1-e4, double check and mate.
First example: 1. e4 e5 2. zBe2 Nf6 3. c3! Be7 4. d4 d6 5. zBd2 Nbd7
Not 3...d5? 4 e:d5 N:d5 5 c4 and wins a piece because d5 is pinned against e8 by the zB on e2.
After these moves, White has some advantage but it is minimized by the way his zB's can't get the open space they need.
Second try: 1. e4 e5 2. d4!? ed4 3. zB:d4 (attacks d8) d6 4. zBe2 Nc6 5. Nf3! (if N:d4 6 Q:d4)
Instead of 5 e4-e5 discovered check, Be6 6 ed6 N:d4 7 Q:d4 Q:d6 8 Q:d6 B:d6 9 zB:e6+, with a very minimal advantage, White looks to get more space and development by offering to trade zB for N.
Third try: 1. e4 e6!? 2. c4!?
The zB seems to need a more open game to be effective, and so Black tries for a closed game with 2 d4 d5; White, of course, resists. Black could try the same thing with 1 e4 e5 2 d4 d6, which might be pretty good.