For my final year, I have to do a project for a module. I want to investigate something in the complex analysis area. I've only covered the basics of analysis, like Cauchy's IT/IF, residue theorem etc. The only thing that's been suggested so far is the mathematics of Aerofoils. Just wondering if anyone has ideas of areas I could look at?
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 always found the Schwarz-Christoffel map to be particularly fascinating and always an area of active interest. The applications are boundless, but really they are used for solving flow-type problems around awful, polygonal boundaries.
The uniformization/Riemann mapping theorem should give you lots to chew on.
A slightly less well-known example is the actual proof of the Runge phenomenon. It is one of my favourite applications of the residue theorem. You can find a sketch of the proof in this document with links to other resources.
In the same vein, Lax and Zalcman's book contains a lot of very interesting examples that one usually do not encounter during a first course in complex analysis.
You may find some inspiration in Needham's Visual Complex Analysis.