In fact, the normal Bishop is a useful piece, but how much more pleasing it would seem to be to use instead the Crooked Bishop.
The Queen then should be a combination of Rook plus Crooked Bishop, and I have named this piece the Crooked Queen.
It seems to me that the 4x16 board naturally limits the scope of chess strategy; centralization means little because you do not have two separate sides to threaten from the center, and alternation (as per Nimzovich) is a strategy that has no place on the 4x16 non-circular board (although on the 4x16 circular board, it can sometimes be applied vertically).
In Race Chess, simply because the Pawns all move clockwise, there is always a duality between defense of your rear and attack against the enemy rear. Of course this means that the game is strategically not so deep as is FIDE Chess, but instead it is as though in FIDE Chess one were to always castle on opposite sides and launch attacks against each others' Kings -- and this is a type of game that everybody likes to play sometimes!
In the traditional 4x16 circular non-race chess, one does have two fronts on which to attack or defend, but because the Pawns face "outwards" they can help in defense. I think that Race Chess, where the Pawns all move clockwise, is a better game because the rearguard Pawns are weak -- it is interesting to try to get value from them!
The question then is whether using the Crooked Bishop and Crooked Queen would improve 4x16 Race Chess. I would say that they do not.
The Crooked pieces add more power to the game, and therefore make the attack more urgent. This is a different game, not necessarily a better one, and there is more risk that somebody might analyze the openings too much and find a way to turn White's initiative into a win.
The current rules of 4x16 Race Chess do have an error in that they require Pawns to circumnavigate the board before promoting, that is, they require 13 advances rather than 6. However, due to the nature of the game, it should usually be over anyway before Pawn promotions happen.
The following UAD shows the opening position of 4x16 Race Chess. Note that part of the board is repeated because doing so makes moves easier to visualize.
a . * P R R P . * . * p r r p . * . * P R b * . P N N P * . * . p n n p * . * . P N c . * P B B P . * . * p b b p . * . * P B d * . P K Q P * . * . p k q p * . * . P K 13 15 1 2 3 4 5 6 7 8 9 11 13 15 14 16 10 12 14 16
This article is a necessary step in the development of 2x2x16 Race Chess.