Final year project ideas - complex analysis

I suggest you to look at some special functions, i.e., gamma function, zeta function and theta function. And the connection between cplx analysis and number theory. It's really interesting that the property of these functions will lead to some property of prime numbers. You can especially focus on the fundamental theorem of prime numbers and the four square theorem.

I would suggest studying Fourier and Laplace transforms. Learning them well will suit you in the future, regardless of which field you end up in.

The theory of minimal surfaces is kind of interesting, there are a lot of nice pictures out there of the strange shapes you can make. Learning about the Poisson kernel and seeing how it works is interesting too.