1st move: 24 pawn moves, 4 elephant moves, 4 knight moves, 2 half-elephant moves. 34 moves possible, compared to 20 in regular chess. vThis will be interesing.
12x12 with new pieces

I am going to post 3 moves by each side at a time except for this time: 5 moves by each side and a diagram. Most of the fighting will be done with pieces, because pawns have to travel very long distances. Maybe I should make pawns able to move farther.
Normally the 'Switching Bishop' (called 'Dragon Horse' in Shogi) is a tiny bit stronger than Rook (on 8x8: R=5, SB=5.25).
The Half Elephant is an interesting piece. Although it should be significantly stronger than a Knight (leapers with 12 targets generally have values just a bit under Rook on 8x8, but the latter of course gets better n a large board), a pair of them still cannot checkmate a bare King. The extra move does not help them to get from c1 to a1 in 3 moves (they are still color alternating), and they still cannot fork a1 + c1 when the King is on b3. So that they can go from c1 to b1 in 3 moves does not help.
[Edit] Sorry, I was wrong about this. Two HE have a quite unsual mate pattern with their King on c3.
The HE at c3 covers a3 here, so that the white King is not needed to close off the a-file, and can be on c3. This forces the bare King to b1, and then 2. HEb4# is checkmate, because the HE can attack a2 and b1 at the same time.
According to the tablebase the end-game is in fact 99.7% won for white (white to move)/

Switching bishop sounds awesome! Wish there were SB'S in original chess. Cool concepts. Is there somewhere you have a beta?
pawn=1 knight=3 bishop=4 rook=6 queen=11 Switching Bishop=6 Half-Elephant=5 Hawk=10 Elephant=10 Superqueen=15
I made the long-range pieces more valuable, and for the hawk, elephant, SQ and queen it is the combined value of the pieces that it can move like +1.
I may play a game against myself with these pieces+board.