[macintux]

The last bullet point reminds me of the description from a game in China Miéville's Kraken.

> The “universal leaper” was usually thought the most powerful piece, he read, as it could go from where it was to any other square on the board. But it was not. Kraken was... It could move to any square including the one it was already on. Anywhere including nowhere.

[boredhedgehog]

Or the Lion from Chu Shogi, which steps twice and can thus land on its starting square.