The Quadlevel version of 3D chess was invented by Robert Koernke Sr. back around 1970, in response to the 3D craze started by a Star Trek episode (This does not resemble Star Trek chess - it has seven boards and some of them movable. Playing this game, since the concepts are the same, will help your 2D game. It's like picking up 3 bats and going to the plate with 1. A booklet written by Robert Koernke Sr. Robert Koernke Sr. is still available at the Ann Arbor University of Michigan Undergraduate Library entitled "The Complete Rules of Three Dimensional Chess" copyright 1970. The only major change in 30 years is the 'Standard' win version. Originally you had to checkmate both Kings, which drags the game out too long. Robert Koernke Sr. and a man named Gerhard Fritsche(Gary) would go to what is called in Ann Arbor, MI the Diag, where the crossroads of the U of M students come together, and play the game in front of a crowd of 50 or so. Recently Gary and Robert Koernke Jr. did that again, with success. Some of the earlier ideas for the name of the game were Bi-sovereign chess; Oligarchy Chess. Robert Koernke Sr. had never given it a distinct name other than 3D Chess. Robert Koernke Jr. went with Quadlevel. Some of the same openings are possible, such as a Queens Gambit Accepted/Declined.
Quadlevel 3D Chess and Checkers