TABLETOP SIMULATOR MOD IS LIVE, COME AND GET IT
I call it Deicide Chess because god is dead and I looked him square in the eye as I pulled the trigger.
Beginning setup looks like this:

That might seem like a lot, because it is! There are a whopping 25 different piece types here, over FOUR TIMES as many as regular chess, and the board is 16x16. That's TWICE the side length and FOUR TIMES the area. However, the goal remains exactly the same: Checkmate your opponent's King before they checkmate yours.
So how does each piece move?
First, some terminology.
The Glossary
Step - Movement to an adjacent square. Notated with a dot in the destination square, or an X for capturing-only steps.
Slide - Movement along a straight path of unoccupied squares to reach a square that is not adjacent. Notated with an arrow that follows the path of squares.
Jump - Movement directly to a square that is not adjacent, ignoring any pieces that may be in the way. Notated with an arrow.
(X, Y) jump - When a piece makes an (X, Y) jump, it moves X squares in one orthogonal direction, then Y squares in a different, perpendicular direction. X and Y can be swapped freely, so an (X, Y) jump is equivalent to a (Y, X) jump.
Orthogonal - Forward, backward or sideways, perpendicular to the edges of the board.
Diagonal - An equal amount of movement along two perpendicular orthogonal directions.
Forward - Towards your opponent's side of the board. (Backward, similarly, is towards your own side of the board.) For White, this would be moving towards the 16th rank of the board, while for Black it would be moving towards the 1st rank.
In any direction - Orthogonally or diagonally.
(Unless otherwise stated, and with the exception of the above definitions, "move" specifically refers to non-capturing moves, while "capture" specifically refers to capturing moves. "Giving check" refers to capturing moves that can only target a King.)
The King

Moves or captures by stepping one square in any direction.
Royal - If your King is threatened with capture, you are in check, and you MUST remove the threat. If you cannot remove the threat, you are checkmated and you lose the game.
Castling - Castling is done by having the King move two squares towards its Rook, then the Rook jumps to the square the King passed through. The King may not castle into, out of or through check, all squares between the King and Rook must be open, and neither the King nor the Rook used may have been moved this game.

The Queen

Moves or captures by sliding any number of squares in any direction.

The Rook

Moves or captures by sliding any number of squares in an orthogonal direction.

The Bishop

Moves or captures by sliding any number of squares in a diagonal direction.

The Knight

Moves or captures by making a (2, 1) jump.

The Pawn

Moves by stepping one square straight forward. Captures by stepping one square diagonally forward.
Head Start - May slide up to four squares forward the first time it moves. (For white Pawns, this only works if done while on the 4th rank, for black Pawns, only from the 13th rank.)
En Passant - If an opposing Pawn uses its Head Start to move past a square where your Pawn could have captured it, you may move your Pawn into that square, then capture the opposing Pawn anyway.
Promotion - A Pawn that reaches the other side of the board (the 16th rank for white Pawns, or the 1st rank for black Pawns) becomes any piece except for a King, another Pawn, a Drunk Pawn, or a Rookie.

The Archbishop

Has all the movement capability of both the Bishop and the Knight.

The Chancellor

Has all the movement capability of both the Rook and the Knight.

The Cannon

Moves like a Rook, but cannot capture like one. Instead, to capture a piece, there must be a hurdle of either color between the Cannon and its target, which the Cannon jumps over to reach the target. The Cannon may not jump if not capturing.

The Archer

Moves like a Bishop, but cannot capture like one. Instead, to capture a piece, there must be a hurdle of either color between the Archer and its target, which the Archer jumps over to reach the target. The Archer may not jump if not capturing.

The Time Machine

Moves and captures like a Queen, but three turns into the future. To do this, declare a valid square that the Time Machine will land on, then remove it from the board with three time counters. Every turn, after you make your move, remove a time counter from the Time Machine, then if there are no time counters left, return it to the board on the square that was declared three turns ago. Any piece in that square is captured, even if it's a friendly piece. The Time Machine threatens the square it's about to land on. Kings who are standing in that square are in check, and if a Sumo Wrestler could push a King onto the square a Time Machine is about to land on on the next turn, the King is in check.

The Champion

Moves and captures by stepping exactly one square orthogonally, or jumping exactly two squares in any direction.

The Wizard

Moves and captures by stepping exactly one square diagonally, or making a (3, 1) jump.

The Kitsune

Moves and captures by making a doubly-bent sliding move; first, the Kitsune steps one square in an orthogonal direction, then turns 45 degrees left or right and slides any number of squares (including 0) in the resulting diagonal direction, then turns 45 degrees left or right again and steps one square in the resulting orthogonal direction. The Kitsune cannot capture until the final orthogonal step.

The Werewolf

Moves and captures by making a doubly-bent sliding move; first, the Werewolf steps one square in a diagonal direction, then turns 45 degrees left or right and slides any number of squares (including 0) in the resulting orthogonal direction, then turns 45 degrees left or right again and steps one square in the resulting diagonal direction. The Werewolf cannot capture until the final diagonal step.

The Centaur

Has all the movement capability of both the Prince and the Knight.

The Fool

The Fool has no movement capability of its own.
Joker - Copies the movement and capturing capabilities of whatever piece your opponent moved last, but not any special abilities, such as the King's Royal ability, or the Pawn's Promotion, except for another Fool's Joker ability.

The Sumo Wrestler

Moves by sliding up to two squares in any direction, but cannot capture normally.
Kimarite - If an enemy piece is in the Sumo Wrestler's path and could otherwise be captured by it, the Sumo Wrestler can instead push that piece one square in the direction in which it is moving. If the Sumo Wrestler is able to push the enemy piece off the edge of the board, the enemy piece is captured.

The Sorceress

Moves by jumping to any unoccupied square. Captures by stepping one square in any direction.

The Rookie

Moves by stepping one square straight forward. Captures by stepping one square horizontally.
Head Start - May slide up to five squares forward the first time it moves. (For white Rookies, this only works if done while on the 3rd rank, for black Rookies, only from the 14th rank.)
En Passant - If an opposing Rookie uses its Head Start to move past a square where your Rookie could have captured it, you may move your Rookie into that square, then capture the opposing Rookie anyway.
Promotion - A Rookie that reaches the other side of the board (the 16th rank for white Rookies, or the 1st rank for black Rookies) becomes any piece except for a King, a Pawn, a Drunk Pawn, or another Rookie.

The Drunk Pawn

Moves by stepping one square diagonally forward. Captures by stepping one square straight forward.
Head Start - May slide up to five squares diagonally forward the first time it moves. (For white Drunk Pawns, this only works if done while on the 3rd rank, for black Drunk Pawns, only from the 14th rank.)
En Passant - If an opposing Drunk Pawn uses its Head Start to move past a square where your Drunk Pawn could have captured it, you may move your Drunk Pawn into that square, then capture the opposing Drunk Pawn anyway.
Promotion - A Drunk Pawn that reaches the other side of the board (the 16th rank for white Drunk Pawns, or the 1st rank for black Drunk Pawns) becomes any piece except for a King, a Pawn, another Drunk Pawn, or a Rookie.

The Prince

Moves and captures by stepping one square in any direction.

The Princess

Moves and captures by jumping up to three squares in any direction.

The Bomber

Moves and captures by stepping one square in any direction.
Bomb - If the Bomber captures or is captured, both the Bomber and the piece it captured or by which it was captured are destroyed.
Big Bang - If one Bomber captures another Bomber, a Big Bang is triggered, destroying pieces in the eight surrounding squares as well.

The Snail

Moves and gives check by stepping one square in any direction.
Snail's Pace - On your opponent's turn after you move the Snail, your opponent may move an additional time. They may not put you in check with any move except the final one made on their turn. This ability does not apply on extra turns, but if you move a Snail on an extra turn, it costs two moves instead of one.
Immortal Nemesis - May only capture the King, and cannot itself be captured.

design notes/thoughts:
- I followed Yuri Chess' lead in ditching the orthogonal pieces that have diagonal equivalents, we've gotta conserve space where we can
- but also everybody needs to have two Time Machines, I mean c'mon
- the Princess felt a bit too OP in such close quarters, and anyway I hear girls can be Princes too
- the Archbishop is flanked by the Archer and the Bishop, that's just basic math
- I wasn't sure why Black would want the snail, since a non-king piece that can only capture kings seems counterproductive to Black's goals; but an uncapturable piece is still potentially valuable for blocking off opposing moves maybe?
- in retrospect I'm not sure what i was doing with the Drunk Pawn, but that feels appropriate; let's say the Drunk Yuri Pawn can move diagonally towards the opposing corner, and capture in the two orthogonal directions adjacent to that diagonal direction? and it gets promoted if it reaches either of the opposing edges? it's sorta like a normal pawn that just didn't notice the board got rotated and is doing its best

(also i'm reducing everyone's Head Start back down to two squares max for the smaller board) - if you use the Fool to copy a King, it can capture other Kings; the Black King-Fool capturing the White King or the White King-Fool capturing the Black King counts as a win for Black, and they also get the Trans Yuri bonus achievement
The next step will of course have to be 5D Yuri Deicide Chess With Multiverse Time Travel

