archknights and archbishops are already invented but i will say it anyway
Invent your own chess piece!

The Crazyknight. Moves and captures two squares sideways and one square forwards, when it gets to the end it becomes a rook.

The Superpawn. The SP moves and captures just like a regular pawn, except that it promotes to a very powerful Reine-Chevalier (RC) (French for Queen-Knight) which, well, moves like a queen and a knight. The superpawn looks like a slightly-taller pawn, and each side has two of them, one each on the c and f files. The RC looks like a knight, but is significantly taller and has a crown on its head.

Enlighten me on archknights and archbishops?
Wow, that crazyknight sounds quite good...much like the higher-class pawn!
The superpawn...always wanted a Rèine-Chevalier...you get those games where it would be checkmate if the queen could also move like a knight and it gets so annoying when it's constantly check but never checkmate...
Fantastic ideas so far!

It's interesting to think about the strategies that using these pieces might entail. For example, your opponent might happily sacrifice a piece to capture one of your superpawns, meaning that you'd want to protect them more than a regular pawn, so you can't just push them to the centre or as part of a chain as a regular pawn. Instead, you'd want to clear out those files quickly so that it can promote and protect the squares around it with pieces.

The Fat Albert ultimate piece. Worth 85 points, it crushes anything in the 63 squares around it and can choose to go to any square it wants. But unfortunately, both sides only get to have 0 of these.
Interesting though they sound, the existence of superpawns would unfortunately wipe out such openings as the King´s Gambit and the Sicilian, as no-one would want to throw one away within the first few moves ...
The Legal Eagle
It has no independant existence but sits (in the tiny shape of a bird of prey) on the King's shoulder and once in each game can be dispatched to attack any piece on the board (of either colour). The opposing player must blindfold himself or herself (justice is blind) and make a dice roll upon the outcome of which either the piece is removed or the corresponding piece (queen for queen, rook for rook etc.) from the other side is removed from the board.

a chameleon pawn. At any of your turns, you can pick any of your pawns and turn it into any other piece, except another king. Or maybe to make it more difficult, that chameleon pawn is chosen at the beginning of the game, maybe an h or a pawn, and that pawn can be turned into any other piece at any time, and if that pawn is captured you have lost an extremely valuable piece.

Somebodysson, what would happen if that pawn got to the end then?
it could change at any move into another piece. Getting to the other end would no longer be a goal of that pawn, because it could change into a queen far earlier than getting to the other side. What could transpire is the game could quickly become a game with two queens per side. It would be more interesting, tho, if on a move the chameleon pawn could become one of your bishops that has been captured, , or a knight giving a treacherous attack via a knight move, or a rook suddenly claiming a rank.
I thought of it because don't you sometimes say to yourself "if only I had my dark squared bishop (or my light squared bishop) now...this would be a way of getting it back.

The Mexican.It moves like a king and cannot be captured nor capture.It's a nice way to block checks.It also revives a friendly piece in its moving range.

But why 'promote' it to a bishop or a rook to claim a coloured square or a file/rank when a queen combines both a rook and a bishop. Promoting to a knight makes sense. Imagine a checkmate by this method of promoting!
yes, good point.
Hello, everyone, just a bit of fun which some of you may be interested in:
Say you were asked to invent a new chess piece for a new variant of chess that chess.com would like to try out. What would this chess piece be?
Here are a couple of questions you could think about whilst inventing the piece:
Tell us, and we'll discuss it and give our own suggestions as well!!
Now, for my own piece...
Introducing the Earl, a piece that can only move one square at the time in any direction (like a King), but can also move to the other side of the board if it reaches the a or h files or the 1 and 8 columns. It's only worth two points, two earls per side, and looks like a crown (no torso!). It can also move right next to the King.
Get the idea? Get writing!
Note: if a post like this has been created in the past, please excuse me, I'm new!!