Zoro is not bowing down to Mihawk to abandon his ambition and dream what lol
The bowing shows that Zoro values his captain more than his own pride and dreams, but it doesn't mean that he's giving up on his dream/ambition in that instance, he's just swallowing his pride.
The only time Zoro actively let go of his ambition was in Thriller Bark, when he was prepared to die to save Luffy. But even that wasn't him giving up on it to save himself/live. It was him saying if he can't even protect his captain, then he's not worthy of WSS anyway.