Everyone's just got to stop translating it as "looking for." There is no implied preposition. The word מחפש means "searching out."
"What are you searching out?"
To have it make sense, flesh out an answer:
"I'm searching out the old water well they said was here, but i can't find it anywhere."
"What are you looking for?" and "what are you searching for?" are both correct and are good translations here. "For what are you looking/searching?" is OK too, if a bit unusual and stiff sounding.
For what it's worth, pretty much all modern style manuals are fine with ending with a preposition. The Chicago Manual of Style calls the prohibition "an ill-founded superstition," and Strunk & White say that "not only is the preposition acceptable at the end, sometimes it is more effective in that spot than anywhere else." The case against it probably started from an attempt to apply Latin syntax to English in the late 17th century.