There should totally be an option for mugging bounties - bounties that are claimed by mugging the target. If people don't understand why this might be fun, it's an especially apt way to punish muggers, rather than hosping, where they can just med out. Also, I don't know about you, but being mugged is a bit more annoying than being hospitalised in my opinion.
There is a clear issue with this which is that targets can see the mugging bounties and be particularly cautious about their money, but forcing targets into some level of paranoia could be an intended effect. Also, there could be an extra fee to hide bounties from the target though this may be a bit too imbalanced.
When it comes to money received from a mug, I personally would be fine with letting the bounty hunter take the money, but perhaps in the future the money mugged from such a bounty would go to the person setting it, though this is not a big deal to me.