The search engine that works behind the scenes to generate the "Questions that may already have your answer" is staggeringly amazing; even this first draft at a not very good title brings up exactly the question I was going to ask as the first result:
On the other hand the search bar doesn't do quite such a good job:
So I'd like to propose that a little of the "New question screen" magic go into the search bar.
Or to propose this in a little less wishy washy way; that (in addition to normal search results) top results from the "Questions that may already have your answer" box go into the search box as if the search had been phrased as a new question.
This will help casual users who may search but may not (attempt to) ask. This will additionally be helpful for experienced users because (due to draft saving) you cannot have two independent new question pages open at once. As such I can't use the new question based search while also drafting a question