I frequently feel sad that complex numbers are not part of the IB SL curriculum. During the time when I taught Algebra 2, it was always my favorite topic to connect algebra, geometry, and the history of numbers all at once.
Here is one cute activity I used to give to my students to illustrate the relationship between complex number operations and coordinate transformations. I recently gave it off-handedly to a student at our school, who was very intrigued by this, so I thought I'd share with those of you who still teach this topic. I think this activity is very eye-opening for the students and very visual, and it gives the various algebra operations a more concrete meaning / some motivation.
I vaguely remember that I had written about this a long time ago, but here I am posting it again since I cannot find the old post. (oops. Lost in the WWW, I guess.)