I believe you could add endless complexity to the game without adding any new piece. Just by giving a new speciall move to any of the existing ones. Just like panws have "en-passant" or kings have Castle, maybe knights or bishops could get a 1 time special move.
Just think of it.
Knights: could get a 2 moves in a row once in the game. As long as they don't take any piece. However thay CAN check with the second move.
Bishops: you can choose one of them and... step to the side! and change colors once in the game. BUT you can't take pieces with that move. And you can only do that while still having both bishops... that would balance the endgame. Since it would be too cheap to switch colors of your last bishop.
Rooks: Maybe they can jump over a piece that is next to them once per rook, as long as the destination square is empty. That could be awesome both for early attack or for versatilty in defence.
the crazy part is.. you could make players CHOOSE what speciall move they want before starting. Knight combo, bishop color, or rook jump. You can't have them all.
I thought about 3D chess - basically, say, three (or more) boards one over another...
...and then there would be a new piece, through which all the others would go up or down :) Let's call it an Elevator.
I didn't make my mind as for the initial position, how many Elevators would each player start with or how they would move (like King?), it's just a germ of an idea. Obviously they'd be of very high value and essential in combinations.
I also thought about "improving" the monarch himself, how about giving the King "two lifes" - it could survive one capture; capturing it would annihilate the attacking piece. That would open lots of tactical possibilities I believe