Let's invent some very weird pieces

https://en.wikipedia.org/wiki/Tenjiku_shogi

Monochromatic Pawn

The Monochromatic Pawn is color bound, and so can only move and capture diagonally forward.  If it is both the Monochromatic Pawns first move and the Monochromatic Pawn is not capturing anything it can move one or two squares diagonally forward, otherwise the Monochromatic Pawn can only move one square diagonally forward.  When the Monochromatic Pawn gets to the end of the board it promotes, but it is only allowed to promote to a color bound piece in order to keep itself color bound.

Also the Dabbabarider

The Dabbabarider moves an even number of squares horizontally or vertically, and can only be blocked by pieces or pawns an even number of squares away from it horizontally or vertically.

See https://en.wikipedia.org/wiki/Dabbaba_(chess)

and also see

https://en.wikipedia.org/wiki/Fairy_chess_piece

 skipping to the part about riders to understand the Dabbabarider.

Hobbit. Moves like Alfil in Shatranj:

But captures like a King:

When Hobbit moves like Alfil, it can leap over the pieces.

AntiCheck. Moves and captures like a queen. But it cannot make moves give a check. First check of this piece must be a mate.

Anti check is contradictory same as conditional check 

About the Mirror.

https://www.chess.com/clubs/forum/view/list-of-pieces-used-for-bulldog-and-infinite-chess?page=4

Princess. Moves and captures like a queen. But it cannot capture the pieces, which are staying near the King.

Pr:f8+ is illegal, Pr:b6 is legal capture.

Bichromatic King. Moves and captures like a King(not royal). But it cannot move and capture from black to black square or from white to white. It must change color. Cannot be checkmated, but can be taken by other pieces.

@ZhenyaChaynikov
Your "Bichromatic King" is basically a piece that moves like a King but Captures like a Wazir.
This piece, and your Hobbit, are quite basic pieces that have movement different from capture. 99% of the invented pieces move how they capture, but when we'd start with movement different from capture, then the possibilities are endless.
One more such piece is the Hunter.

@ZhenyaChaynikov
Cat and Mouse have been invented with different rules by @captaintugwash.
The Charriot is a name used for more pieces in Asian variants. The Princess has been used as name for various pieces. You can check here if your name for a piece has already been used. I myself prefer to avoid that.

Fisherman: moves like an Amazon (queen plus knight), but can only capture pieces who are not adjacent (orthogonally or diagonally) to any friendly pieces

Dude I am sorry, I have been really inactive lately we can start the game on Saturday perhaps?

Sure, can you create the thread? I can post diagrams.

Stain: moves like a bishop. Any enemy piece that moves through the path it took in its most recent move is turned into a pawn.

Morfer. A piece that moves like a king, but if / as long as it is blocked by friendly pieces in all directions, only then it gets jumping abilities.

Baron. Piece that moves and captures like a king but pawns can not capture it.

Mermaid. Moves like a queen, but can jump over pieces, takes like king. Lots of tactics, it's also very good at helping with a mate.

A lamp. Moves 1-2 squares in any direction. It does not capture, but can highlight the pieces standing on adjacent cells with it (after it s move). Pieces, illuminated by the lamp, should be randomly scattered on any free squares, but so that it does not lead to the check's own king.

Randomly?

The Instigator. The Instigator moves like a king normally, but if it is a queen move away from an enemy piece that is in line with another enemy piece, (Example: Instigator on e3, enemy Bishop on e6, enemy Knight on d5) then the Instigator can "agitate" the enemy piece. (The Bishop) Once a piece is agitated, it will capture the nearest piece on the next turn. (The Knight) If there are no pieces in is line of sight for 4 moves, the piece loses its agitation. 2 rules for this: The King and Queen cannot be agitated, and more than one piece can be agitated at a time.

Alternatively, the Puppeteer. The Puppeteer moves like a queen WITH THE EXCEPTION that it cannot move more than 5 spaces in any direction. If the Puppeteer captures a piece, that piece becomes its "puppet" and is placed adjacent to the Puppeteer. Puppets move the same way they would normally, but they attack enemy pieces. If the Puppeteer is captured, then all of the "puppets" (there can be more than one at a time) return to normal.