This is to some extent similar to answer to a related feature request on Meta Stack Exchange.
Since the featured ("bountied") questions are already shown relatively prominently, I do not think that adding a separate spot for them is needed.
- When a bounty is started, the question is bumped.
- These questions are shown on a separate tab.
These questions are visually distinguished in question lists (blue square showing the bounty amount).
In particular, fixing constantly a top-spot in some of the tabs for a question with a bounty seems like too much attention to featured posts to me.
Probably a better thing to do if you think that bounties to do not get enough attention could be making more users aware of the existence of the featured tab. Although this is probably not an easy thing to do, there are some reasonable options which might help a bit. For example, if you see an interesting featured question, you might mention it in chat. (The same goes for any interesting question, regardless of bounties - and this might also increase awareness of the MathOverflow chatroom.) Some possible downsides of bounties and reasons why some user dislike them were mentioned in the comments - perhaps a separate discussion about pros and cons of bounties and how they can be used in a way useful for the site would be interesting and as a side affect it might attract more users to featured posts.
It is also possible that some users will add featured questions to their custom question lists - this is a new feature which is currently being tested on a few Stack Exchange sites, but this new functionality should be available on all sites in the network at some point during this year. So this might also help the questions with bounties to get a bit more exposure.
Anyway, since we are discussing bounties, let me leave a few related comments.
- Several users on MathOverflow expressed they dislike of reputation (and badges). So it's possible that some users can be put off by a bounty - since they do not want to make an impression that they are answering just because of the reputation points. (Although I only remember one case where removal of a bounty by a user who would receive it was requested. It was on Mathematics, but the user in question is probably familiar to many MathOverflow users.)
- I would agree that a bounty on a question can indicate that it is an interesting question. The most frequent reasons that somebody offers a bounty seem to be: 1. The asker received no satisfactory answer for some time. 2. A user considers someone else's question interesting and wants attract more attention to it. 3. A user considers some answer exemplary and wants to reward the answerer. Each of these reasons correlate, to some extent, with some level of difficulty of interestingness of the question.
- Information that somebody offered a bounty does not go away after the bounty period ended. So it is possible that some users might use this information to find questions which had bounties. In particular, in the users profile there is bounties tab which shows both offered and earned bounties by a specific users. (And it would not surprise me if somebody occasionally looked at profile page of another user who has similar interests or who is a well-known expert in some area - to see which questions they asked, answered or even where they offered a bounty.)
- It is possible to find past bounties using Data Explorer (SEDE). Some queries which can be used for this purpose are listed in this answer.
- As an experiment, some time ago I started a chatroom called Listing bounties. (Partially in connection with my post: What would be good place to list unresolved bounties?) This room should contain most (almost all) featured questions since the time it was started. The intention was to make the bounties easier to search and shown in a more user friendly format. (In chat, you can see a preview of the question. You can easily search for past bounties in a specific tag or with the title containing some keyword.) Of course, one can imagine also various other technical solutions to create some list of past bounties where the format would be more user-friendly and which would be easier to search than the results from SEDE.
- The observation (made in the comments) that many of the bounties are offered by (relatively) low rep users to attract attention to their post is definitely correct. However, bounties can also be used to make the best content of the site more visible. (I have seen some bounties on MathOverflow which were offered to reward an existing answer - which seems to be in the similar spirit. Of course, users who have more reputation have wider possibilities to do something in this direction.) I will mention that on Mathematics site some moderators and experienced users have recently discussed the use of bounties to highlight the site's best contest - as you can imagine, on a high traffic site many good posts can go unnoticed. There are plans to start a separate chatroom devoted to this and it should probably be launched at some point during this year. (When it is created, I'll include a link in the comments.)