Let's invent some very weird pieces

What happens if your King is in check and your Cataphract is also 'in check' at the same time?

Any piece that captures 3 pieces becomes a jumping variant- It cannot capture & jumps over when it tries. This applies to kings & pawns, & when a jumping pawn promotes, it promotes into a jumping variant. The jumping pieces can still deliver checkmate & checks however, and can threaten kings when blocked as a result. Also knights have a second jump over their first change in direction (Only if capture of course.) Pawns still are blocked.

the pumpkin. No matter where it is on the board, the pumkin can always move to e6, h2, b5, and g3. It can not move to those four squares if your piece is occupying it, but if you opponent occupies it then moving to e6 h2 b5 or g3 would be a capture

The weirdest piece I invented for one of my own Chess variants is the 'Werewolf' of Werewolf Chess. It slides like a Queen, but maximally upto 3 squares, but it can jump to the second square (but not the third) directly, even if the first square in that direction is occupied. And when it jumpsover an enemy piece that way it can (but doesn't have to) remove it in the process (like a Checker capture), even when it also captures something on its target square (so it is capable of double capture). But that is not yet the strangest property: every piece other than a King that captures it, will instantly promote to a Werewolf. (Which effectively prevents it is traded out of the game early.) That idea of a 'contageous' piece was taken from Maka Dai Dai Shogi, btw.

Another pretty weird piece, which I did not invent, but stole from Chu Shogi to feature it in my own variant Mighty-Lion Chess: it moves as a King, but twice per turn, and doesn't have to stop after a capture. (So it can do hit-and-run, and double captures, and rifle captures on adjacent squares.) And when it captures another Lion it becomes absolutely royal for one turn. (Implying you cannot capture an enemy Lion that is protected with it, unless it is through hit and run.)

Knight-relay Chess has pieces that acquire extra moves when 'seen' by a friendly Knight; Chameleons acuire the moves of a piece that sees them, and can use them to capture that piece.

'Bifurcators' are sliders whose moves are 'deflected' by the presence of other pieces. E.g. one can start moving as a Rook, but when running into the first friendly piece change direction just before it, to continue sliding like a Bishop. (An enemy piece would siply be captured.) I used Bifurcating Rooks and Bishops in my variant 'Bifurcator Chess', but it was more to demonstrate a possibility, and I was not very happy with the result. (Too complex.)

The idea of having one piece promote if another one is captured is interesting. It would be another method to prevent a piece is easily traded out of a variant that would degenerate to orthodox Chess after the piece gets traded. You could for instance stipulate that the most left Pawn (the most backward one if it is doubled) turns into the piece when the original piece gets captured.

For testing purposes I sometimes use a piece I called (Lion) Cub: it moves and captures like a Knight, butin addition it can rifle-capture like a King. So when you end up next to it, you are cooked.

That reminds me of the Fire Demon of Tenjiku Shogi. Not only does it have a strange move (like Bishop or sideway Rook, but also up to 3 King steps in independently chosen directions, stopping at the first capture), but it 'burns' every enemy on squares next to the one it lands on. (Meaning it can capture up to 8 pieces per turn.) And this burning power is also passive: every enemy that ends next to a Fire Demon is incinerated, and instantly disappears. (They can capture it by landing on it, though.) Even an enemy Fire Demon is fried this way, before it can do its own roasting.

The 'jumping generals' of Tenjiku Shogi are also pretty weird: they move as normal sliders, but can in addition jump over an arbitrary number of pieces (friend or foe) to capture something on the square they land on. They cannot jump over each other, though, or capture a King that way. (In fact they are 'ranked' 1-3, and can jump over generals of lower rank or normal pieces.)