Bounty-hunters can already go and find questions that have a bounty by clicking on the featured questions tab -- surely that's even easier than randomly happening across someone's signature.
This seems like extraneous information that doesn't add meaningful value. That is, bounty hunters aren't likely to care about the user they're hunting for -- and if they are they can see that information attached to the question on the featured tab. So adding this information to the signature just takes up some more space.
Besides the point, indicating a user has opened a bounty still doesn't reveal the question they've opened the bounty on, so any viewer is still a click away.
On the flip side, I can't imagine any benefit to a user advertising they've opened a bounty, opening a bounty already advertises their question in a reasonable way.