I found an interesting variant called "coin chess":

On the square e3 there is initially placed a coin. The coin moves always like the moving piece, with the same direction and distance. The coin must not move onto an occupied square or off the board, but may leap over pieces.

The player to move moves the coin first, then the piece. E.g. white can open the game with e3-e4, e2-e3, but the coin can't be placed onto the origin square of the moving piece.

Winning is by checkmating or stalemating the opponent.

Edit:

1) the coin should initially be on e4, otherwise White has a forced win e.g.: 1.e2-e3 e7-e5 2. Ng1-e2

2) there are two ways to handle check/checkmate: a) ignore the coin, e.g. if white has Re1 and black Ke8 and the coin is in the e-column and no other piece in between, black is in check b) don't recognize check; winning is by capturing the king or stalemating the opponent. Then in the example black is not in danger, white can't move e1-e8 because the coin would leave the board.