I'm wondering about two courses, complex variables (applied complex analysis) and partial differential equations. Do they increase mathematical maturity, and when is the best time to take them?
Complex variables has meaningful application in many other courses, on the other hand, while PDEs come up in much of differential geometry etc. it is usually the case that the methods needed to solve that PDE are specific to the world in which it arose. In other words, take complex first. However, if only one course is available, I'd take whichever I could. All of this said, it really depends a bit on your target for your education (pure vs. applied etc.)
In your comment you asked
why isn't complex variables, partial differential equations, required for a pure mathematics major? according to "What is the ideal course sequence for an advanced student of mathematics?"applied/computational ODE/PDE is optional. They didn't even say differential geometry was required and complex variables wasn't required either. How can a mathematician not be required to know complex variables and pdes in computation?
Unfortunately, we typically have degrees which do not "require" that much. Or, perhaps it's fortunate, I know the classes I have taken by students who are not required to be there tend to be more studious and serious about learning. It's a funny thing, but, writing a law need not change behavior. In the current university setting we have mostly students who are coming to college to get a job afterward. So, the idea of taking course work which is not somehow directly relevant to their job is frowned upon. Ironically, in this same system, we drown out their credit hours with courses like "getting to know the university 101" or "how to write" or "how to read" or "how to think" or "how to use a spreadsheet" or well, you get the idea. A typical DCP has what, maybe 40-50 hours of math and 70-80 hours of everything else under the sun.
The truth is that ought to flip for students who intend math as a career. You probably need to study 80 hours of math just to get a really good sense of what math is and what is basic and known. This complex variables course and this PDEs course are honestly just introductory. You're learning what has been known for about 200 years. To get up to the present time, there is a lot you need to cover.
But, I still don't know why you're asking these questions, so, my advice may be far off base. I dream of a world where most Mathematicians have some idea about what their peers in other fields are trying to accomplish.
It is not clear to me if all the predicates of that question apply to you...err...I mean a typical student. [Know you want to be a mathematician, only know algebra and trig so far.]
Basically, I don't think it matters a huge amount, but if you...err...the hypothetical student is more to the applied side (as you note), then PDEs has a little more bang for the buck than contour integration. Definitely if you are an engineer or scientist. Also, a lot of good books are set up that way already (Kreyszig Adv Engine Math, Weinberger PDE).
But it probably doesn't matter much for applied courses. Neither course is really a pre-req for the other.