"Why do we need to study numbers which do not belong to the real world?"
I don't think you can answer this in a single class. The best answer I can come up with is to show how complicated calculus problems can be solved easily using complex analysis.
As an example, I bet most of your students hated solving the problem $\int e^{-x}cos(x) dx.$ Solve it for them the way they learned it in calculus, by repeated integration by parts and then by $\int e^{-x}cos(x) dx=\Re \int e^{-x(1-i)}dx.$ They should notice how much easier it was to use complex analysis. If you do this enough they might come to appreciate numbers that do not belong to the real world.
