Over 11 years ago, there was a prior feature request for "bounty hunter" badges, which was received with much acclaim (170+ votes). The request was declined, however. I want to put forward a case for considering this issue again.
Bounties were literally what got me to start contributing to SE in the first place: I had a question that went unanswered, and a desire to put a bounty on it was my first motivation to answer questions so that I could get enough rep to fund the bounty. In this way, bounties are in some sense the only thing that gives value to rep. It thus is in the interest of the entire SE community to provide strong encouragement for users to answer bounties, as this enhances the value of all SE contributions. We want to make posting bounties a worthwhile investment, and for that we need to make sure that we are encouraging good answers to bounties.
Currently, for the sites on which I am active, I don't feel like answering bounties is the best investment of time for most users. My answers to bountied questions rarely get more than 3 upvotes, which I guess is a reflection of how many users are scrolling through the bounty page (not many). Compared to answering new questions or hot questions, this is not the best way to earn rep, even if you do get the bounty. This is even more true when one only gets the auto-awarded bounty (from when the person offering the bounty forgets about it), which is typically only a paltry 25 rep. I think it is totally reasonable to provide an extra incentive for pursuing bounties. It would not be excessive.
A final reason why I think bounty hunter badges would be a good idea is that bountied questions are themselves often more subtle or difficult than normal questions. The typical bountied question (at least for the sites I am active on) is a good question which has received no answers either because it is difficult or because it is highly specialized. I personally love these sorts of questions, but I don't feel like the reward is proportional to the amount of effort required. Answering these sorts of questions is an art, and I feel like it deserves recognition in the same way that tag badges recognize expertise in subject area.