How are Complex Numbers and Complex Analysis used in CS?
I haven't seen them pop up a lot in "Computer Science" per say but there are quite a few closely related fields where they are very present.

I can think of signal processing for example, where Fourier transforms and other complex transformations are fundamental tools. Designing filters for signals also often boils down to placing poles in the complex plane. 

In  control theory and the analysis of complex systems you also sometimes analyse the stability of systems by looking at complex poles of algebraic fractions. 

I imagine complex numbers must pup op a lot in quantum computing also, as a pure quantum state is represented as a complex vector in some basis.

I am only passingly familiar with the fields I mentioned so there might be a lot more ways complex numbers are used in them, and also a lot more fields I am not familiar with.