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.