Tristan Needham's book Visual Complex Analysis is full of these sorts of gems. One of my favorites is the proof using complex numbers that if you put squares on the sides of a quadralateral, the lines connecting opposite centers will be perpendicular and of the same length. After proving this with complex numbers, he outlines a proof without them that is much longer.
The relevant pages are on Google books: http://books.google.com/books?id=ogz5FjmiqlQC&lpg=PP1&dq=visual%20complex%20analysis&pg=PA16#v=onepage&q&f=false
