I recently answered a bountied question and my answer was accepted. I got the 50 rep points for the bounty but was surprised there was no badge/medal for collecting my first bounty. Considering you get badges for casting your first vote, answering an old question, and even just visiting the site every day for a month, I found it very surprising there was no badge for successfully collecting my first bounty. Since I assume that badges exist to encourage users to engage in a variety of desired behaviors, it seems there should be one for collecting a first bounty. Is there a way to propose that a “bounty hunter” badge be created?
Badges are shared through the entire StackExchange system. The best place to ask this is on the StackExchange Meta.
Bounty related badges have already been suggested, in 2009, so I wouldn't suggest holding your breath. They don't seem to be high on the priority list.