Firefox has a bug where form text/texterea inputs in a DIV that overlaps a DIV with position "fixed" don't show the cursor.
In the case of Complete Search, the query input is part of the DIV "left" which has z-index greater than the z-index of DIV "left_bg" (the background DIV of "left"). That means "left" overlaps "left_bg".
Workaround: for Firefox set the position style attribute of DIV "left_bg" to "absolute", when the query input field get the focus and set it to "fixed" again, when the query field lose the cursor.