Autocomplete doesn't work well inside fixed position divs

If you have a div with "position:fixed" and an input tag inside, the autocomplete menu will scroll with the rest of the page. Instead, it should stay with the input tag.

The example I have doesn't look very pretty but you can see it in action here: http://jsfiddle.net/mkVje/

Try typing something in the input box to get the drop down. Then scroll the window down and you'll see the drop down move out of the way. If you were already scrolled down on the page, the search suggestions appear in the right place, but once you scroll some more it moves again.