Interesting!
Variant Suggestion - Alice Chess

I initially thought this is just another variant of 4 players chess.
How about the variant with a 16x8 board (with a normal set of pieces on each 8x8 half of the board)? So the pieces can move on the entire 16x8 board, transferring from one half of the board to another. Seems like an alternative to the normal 4 players chess board (but I assume, the neighboring sets of pieces has to be allies, otherwise the game could quite be imbalanced by the stuff like Rh1 (board A) captures Ra1 (board B) on move 1).

I hate this kind of nomenclature, even the name fairy chess irritates me, just tell me if a real life archbishop is a fairy please! this kind of names demeans chess variants and in this case also infantilizes. Could it be called Wonder chess, mirror chess, reflex chess? sure! But lets call it Alice instead.
About the variant it's intresting, you could also say that it is some kind of 3d chess on an 8x8x2 board where pieces move vertically and capture horizontally.
Alice chess - Wikipedia
This seems like a very interesting variant