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The Piececlopedia is intended as a scholarly reference concerning the history and naming conventions of pieces used in Chess variants. But it is not a set of standards concerning what you must call pieces in newly invented games.

Piececlopedia: Rabbit

Historical notes

The Rabbit is a doubly-bent rider, inspired by the Gryphon and Aanca, which are only singly-bent riders. It was invented in September 2004.

Movement

The Rabbit makes a knights leap (this move cannot be used to capture, and the Rabbit cannot stop yet), then

1) Turns 45 degrees and continues as a nightrider, turns 45 degrees either way and makes a knight jump again, finishing it's move (It must finish the move with this knight leap, it cannot stop or capture with it's nightrider move)

2) Turns 90 degrees and makes another knight's leap, i.e. it makes a nightrider move of zero steps then proceeds normally

Since it cannot stop until it's final knight leap, it can be considered a lame piece. On the other hand, it has a wide range of squares it can end up on in just one move, and does not have the problem the Gryphon has, being that it cannot retreat the same way it moved in the first place.

Its move requires a lot of space to become effective, thus will only be useful for variants with a large board (or a wrapping board?). At the same time, it is rather confusing to visualize.

Movement Diagram

Since the Rabbit's move is too drawn out, a 16x16 board has been used to better illustrate it. 
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   | . |   | X |   |   |   |   |   | X |   | . |   |   | X |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   | X |   |   |   |   | W |   | W |   |   |   |   | X |   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| . |   |   | W |   | X |   |   |   | X |   | W |   |   | . |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   | W |   |   | X |   | . |   | X |   |   | W |   |   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| X |   |   |   | . |   | X |   | X |   | . |   |   |   | X |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   | X | X |   |   | . |   | . |   |   | X | X |   |   | . |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   | W |   |   | X | . |   |   |   | . | X |   |   | W |   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   |   | . |   |   |   | R |   |   |   | . |   |   |   | X |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   | W |   |   | X | . |   |   |   | . | X |   |   | W |   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   | X | X |   |   | . |   | . |   |   | X | X |   |   | . |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| X |   |   |   | . |   | X |   | X |   | . |   |   |   | X |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   | W |   |   | X |   | . |   | X |   |   | W |   |   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| . |   |   | W |   | X |   |   |   | X |   | W |   |   | . |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   | X |   |   |   |   | W |   | W |   |   |   |   | X |   |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
|   |   | . |   | X |   |   |   |   |   | X |   | . |   |   | X |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+
| X |   |   |   |   | . |   | X |   | . |   |   |   |   | X |   |
+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+---+

And here's an illustration of the Rabbit's move in one direction, on a 10x8 board: 

+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   |   | X |
+---+---+---+---+---+---+---+---+
|   |   |   |   |   | X |   |   |
+---+---+---+---+---+---+---+---+
|   |   |   | X |   |   | . |   |
+---+---+---+---+---+---+---+---+
| R |   |   |   | . |   |   |   |
+---+---+---+---+---+---+---+---+
|   |   | . |   |   |   | X |   |
+---+---+---+---+---+---+---+---+
|   |   |   |   | X |   |   |   |
+---+---+---+---+---+---+---+---+
|   | X |   | . |   |   |   |   |
+---+---+---+---+---+---+---+---+
|   |   |   |   |   | X |   |   |
+---+---+---+---+---+---+---+---+
|   |   | X |   | . |   |   |   |
+---+---+---+---+---+---+---+---+
|   |   |   |   |   |   | X |   | 
+---+---+---+---+---+---+---+---+

(R is the Rabbit, dots (.) are squares that may be stepped on as it moves, X's are final destination squares, W's can be ended on or can be passed through mid-move!)