On an 8x8 board there is not much difference between the two pieces. If you want to see an example of this ending from a practical game, visit Jocly.com and check out the most recent game that Panzerschiff and Musketeer Chess have finished. If I have a chance later in the weekend, I will see about posting a diagram and perhaps a few brief comments on that ending.
musketeer endings 2, Archbishop vs chancellor
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The above endgame position arose in a match game between Zied Haddad (w) and John Vehre. (B) White just played 36 Ag5-d2 in response to black's 35....Cd7-d5. With the Rooks opposing each other the game swiftly went into a pure Chancellor vs. Archbishop endgame. Optically black stands well. He has the nominal material advantage of a Chancellor vs. an Archbishop and his pawn structure is sounder. However, an Archbishop is very agile in this type of semi closed position and white pretty easily held a draw in this game. Play now continued. 36...Kc6 [ Black probably should have considered 36...Cf5 to keep the Archbishop from penetrating into his kingside. However instead, I thought improving my king for a possible rook or pawn ending was the approach to take.] 37. a4 e5 38. Ah6 f6 [38...f5 was my original intention, but arriving here, I did not like 39. Re1 Cc3 40. Rxe5 Cxc2 41 Af7 Rxd3 42 42 Ae8+ Kb7 (not 42...Rd7 43 Re7 Cd4 44 Rxd7 Cxd7 45 Kg3 and a position worth remembering. The Chancellor can never escape this pin!) 43. Re7+ Kc8 44 Ac6 and white at least has a draw here.] 39. Ag7 Cd6 40. Kf1 b5 41. ab5+ Kxb5 42. Ke2 Rxe4+ [Black isn't making any progress either with 42...Kb4 43 Kd2 Rxe4 44 fe4 Cd7 45 Ae8 or 44....a5 45 Af8.] 43 fe4 Cd4+ 44 Kd1 Cf3 45 Ae8+ Kb4 46. Ac6+ Ka3 [46....Kc3 47 Aa4+ again is a draw although white might consider playing for more with 47....Kb4 48 c3+ and then capturing on c5. ] 47. Ab5+ Kb4 Draw
One would expect a Chancellor to be worth 8 pawns and some change and the Archbishop 7+ as well. But Kings do like to hide in corners in a normal chess game, but that is just the place where an Archbishop would want them to be. Castled kings on g1 or h1 can be mated by an Archbishop on f3. The piece may have intrinsic value beyond straight mathematical formulas.
The Chancellor stands on d6 on the 42nd move so 43...Chf4+ would be an illegal move in that position.
A chancellor does not have a diagonal move, instead combining the move of a Rook and a Knight. d6-f4 would be a 2-square move on the dark diagonal. Also at the end of this ending note that 47 Ke2? [Instead of 47 Ab5+] Cd4+ 48 Axd4 cd4 would have led to a winning pawn ending for black. What black had been playing for with his king excursion.
The values of Archbishop and Chancellor have been very well measured for the purpose of Capablanca/Gothic and Seirawan Chess. This showed that C is worth about a quarter Pawn more than A, while Q is worth about half a Pawn more than C. E.g. when you play a Seirawan match where one player starts with two C ('Elephants' in S-Chess terminology), and the other two A ('Hawks'), the side with the C will do better (e.g. score around 59%). But if you then handicap it with Pawn odds, the player with two A and the extra Pawn will have an approximately equal advantage. So CC is about halfway between AA and AAP.
In 'regular' Chess starting one player with C instead of Q is already enough to reduce its score to ~41%, but giving him an extra Pawn also almost exactly flips the advantage. So Q is halfway between C and CP, and a C-Q disadvantage can be almost exactly canceled by the advantage of having a Bishop pair (which also is half a Pawn).
The chancellor is superior but lets see it