Amongst the reasons for starting a bounty, one can select:
Reward existing answer
One or more of the answers is exemplary and worthy of an additional bounty.
Why not enable one to reward excellent questions in a similar manner?
It was recently asked if certain sites could be changed to award 10 rep points per question upvote instead of 5. The response was that it wouldn't really be possible to implement that change due to how the SE platform is designed.
But Jaydles acknowledged that on certain sites the questions actually provide more value than the answers:
I do agree with the underlying premise here: On many of our smaller sites, good questions are the single most important bottleneck to growing stronger communities. And on a few more specialized sites, like Code Golf, questions actually take more work, and the askers are generating more value for the answerers than vice versa, which is the opposite of the norm.
So I think that since questions seem to generally get fewer upvotes than answers (as well as those upvotes being worth half as much as answer upvotes), there should be ways to award exemplary questions. Being able to award bounties for those questions is one way of doing that. It would help to keep people motivated to ask great questions that generate much of the value on certain sites.
I think this suggestion might be particularly valuable for recognizing great questions in less-popular niche topics. If I'm a FooBar expert, and there aren't too many FooBar questions, and then somebody asks a really good one, then I might be able to answer it - but me answering it won't get the question much attention. Even placing a bounty might not net much rep for OP.
I definitely like the idea of being able to reward exceptional questions out of my own rep. Rewarding great questions is something Stack Exchange explicitly encourages - by voting. But voting doesn't give a complete picture, since it relies on attaining some critical mass of popularity (hence many "hot", popular, and highly-voted questions are de facto not too great). Personal rewards for exemplary questions, just like for exemplary answers, sound like a nice way for a passionate individual to take responsibility for some of that encouragement.
However, I confess I don't know how widespread practical use might be, and there are certainly concerns such as gaming the rep system, or (maybe?) jump-starting a promising new user too quickly.
The problem with offering bounties on questions is it doesn't encourage any behavior from other users. When users see a bounty available on a question, it encourages them to provide a detailed answer in an effort to be rewarded the bounty. Even if you are "rewarding an existing answer" it is still possible that someone else may find something even more useful than the answer you originally wanted to give the bounty to, which is one of the reasons why the time limit still exists.
The point here is always to get the best answers that you can, not to artificially discourage people from answering. - The Establishment
If you start a bounty, part of your contract with the community is to allow everyone to have a shot at potentially earning it with a great answer. - Jeff Atwood
If you're awarding the bounty to a question, there is no competition. It's not possible for someone to "ask another question" that competes with that one in order to gain the bounty (nor would that be helpful to the community). There's a single question that will receive the bounty, no matter what. So now you have this bounty that has to be awarded to the question's creator and people look at it like "who cares, we can't be rewarded with it so why continue reading?"
A bounty is a special reputation award given to answers. This feature was designed to motivate answerers, and help questions get the answers they deserve.
Ultimately, questions just aren't worthy of bounties. They're questions. They don't provide any factual information that would help others in the future. They're merely details to help others answer the question and a path for other users to find those provided answers, which are actually what helps them. This not-as-importance is already expressed by the half-gain of reputation on questions.
The traditional reward for asking a good question is getting a good answer.
Rep and such are icing on the proverbial cake, but pale in comparison to that primary reward of a solution to a practical question based on a real problem.
Well... I applaud the cleverness of the folks who realized they could use this software in strange and delightful ways that the creators never envisioned much less designed for. I wish 'em all the best. But this is still a worthless feature for the vast, vast majority of people using these sites, a large increase in complexity for precious little real value.
A bounty could/should upon expiration be automatically awarded to the question.
This sure is where I'd like my about-to-expire bounty to go, rather than dissipating into the void just because no new answers were rushed into posting.
A bounty certainly does reflect thoughtful put-your-money-where-your-mouth-is appreciation of a question.
Waiting until expiration would not dilute the incentive for new answers.
(To avert an uncontrolled feedback loop, a bounty started by a question's own poster would be voided upon expiration, just as now.)
Gaming the system ?
I think I find it much easier to judge the quality of an answer (Does it work ? Is it understandable and straightforward ? What further information do I gain if I assume that the answer is really correct?) than the quality of a question, particularly if it comes from an area where I do not have enough knowledge. So how can I argument that this question is really a good question ? Especially if we have a vote ring which rewards its members ? When they defend themselves with "But I really do find this question excellent!" ? How do you argue against that ?