Almost never ... only when the song is actually IN C, or there is a fingering anomaly that is better facilitated by the hand positions in another key. Why "C" anyway? It's really not the easiest key to play in? It's easy for newbies because they don't need to deal with accidentals, but once you know music ... C is just another key.
Playing in the same key can make your chord voicings sound stale of you're not careful. What happened to all those years on accordion? Isn't that supposed to be the "theorist's" instrument? (he said with love)