The BD
The BD is a FIDE Bishop to which is added the power of jumping two squares up, down, left, or right (the two square Rookwise jumping piece called the Dabaaba is known from ancient chess history). For example, from e4 this piece can go to d5, and if d5 is empty can continue to c6, b7, a8; or it can jump to e6, even if e5 is occupied; or it can go f5, g6, h7, jump g4, slide to f3, g2, h1, jump e2, slide d3, c2, b1, or jump c4.To clarify things, let me say that "When the BD makes a Bishop move, it does not jump over other pieces, it really makes a normal FIDE Bishop move. Only when the BD moves two squares Rookwise can it jump over obstacles."
Notice that this piece is colorbound. If it starts the game on a White square, it will never see a Black square all game long.
This piece is usually a little bit stronger than a FIDE Rook, for most of the game, in most positions; and potentially noticeably weaker than the FIDE Rook in some late endgames.
I usually use a physical Rook to represent this piece on an ordinary board with an ordinary set of pieces (this piece has a value of Rook), but I can understand if you prefer to use a Bishop.