http://www.superchess.nl/indexengels.htm
This piece is called the counsellor.
But note that Superchess assigns rather uncommon names to unorthodox pieces, completely ignoring the often centuries old historic names of such pieces, and the names under which they are commonly known in the hundreds of other variants.
The name Archbishop is commonly associated with the B+N compound (but called Princess in Superchess). The B+K compound is usually referred to by its Shogi (Japanese Chess) name 'Dragon Horse'. In a FIDE context its value is indeed slightly higher than Rook (about a quarter Pawn).
If the bishop could also move as a king (except castling, of course :) ), there would be some changes in the game:
- mate with king and bishop against king becomes possible
- bishop doubling on diagonal, similar to rock doubling
- bishop and pawn against king: pawn can be always promoted without help form king
- totally different bishop endgames (due to color changes)
Practically, the Bishop's value would be similar to the Rock.
The basic idea behind the modification is to speed up endgames and bring in more tactics, while keeping changes at minimum level. So it can be played with a standard chess set and default start position.
I have some ideas about boosting pawns, kings, and knights, but let's focus on bishops here. Please, share your opignion.
In the position below, after Be7 (or Bc5-c6+) the white pawn will be promoted in a few steps: