"Why Do I Need Complex Numbers?"
I have heard students ask this question over and over in one form or another. A recent incarnation was featured in this video by 3Blue1Brown. I think this version strikes at the heart of the vendetta that many students have against complex numbers. "What is an application that is impossible to achieve without complex numbers? Convince me that they are needed. They are fun to work with, and maybe they make things easier, but we can do without them." (Edited for grammar and spelling.) At the heart of this question is a reasonable request. When introduced to a new mathematical concept, we should demand reasons, even if the answer is necessarily a bit delayed. High school students in the US are often introduced to complex numbers with little motivation. To them, this new number system can seem like a lot of complexity with little payoff. Maybe a few applications are presented, but one might argue they could be achieved without complex numbers, even if it requires sacr