Not a very mathematical answer (sorry), but if you're looking for something visually interesting to introduce complex numbers, M. C. Escher has got you covered. I won't try to explain it, because I'll do a bad job. I'll let the real mathematicians do that:
edit: see also: http://www.ams.org/notices/200304/fea-escher.pdf
http://www.math.uconn.edu/~kconrad/blurbs/grouptheory/CstarqZ.pdf
and also this youtube video: http://www.youtube.com/watch?v=9WHdyG9mJaI