In physics and many other graduate engineering courses, PDEs are a pre-requisite to studying things like harmonic functions, elasticity, hydrodynamics, EM and electrodynamics, etc. Also, you will need a healthy dose of vector calculus and complex analysis.
There is not a certain order to learn topics. The usual things to learn around the same time as elementary partial differential equations (and covered in the same books) are ordinary differential equations, linear algebra, vector/tensor calculus, calculus of variations, complex variables, integral transforms, probability/statistics, and numerical analysis. One always seems to have more mathematics to learn than time to learn it.
I can't speak to engineering too much, but from a physics point of view any comprehensive math course is a useful math course. Even things like number theory find themselves popping up in research.
That being said, some of the key starting points are linear algebra, differential equations (ordinary and partial), and complex variables.
Basically piggybacking off of lurflurf "One always seems to have more mathematics to learn than time to learn it."
Depends on how complex is taught. If you see it in a sort of "Math Methods" kind of class, it could be a lot of fun. Doing problems like contour integration and such.
If you take a math majors complex analysis, it is going to be a stranger version of real analysis which is all proof based.
Now some find doing proofs hard, some find it challenging but fun.
Just check your course roster for physics/math. Some schools offer both a pure complex analysis class and an applied complex analysis class whereas others might only offer one or the other. You would have to choose based on your needs/requirements and interests, amongst other things. For example here are the course descriptions for the introductory pure and applied complex analysis classes at my university:
"MATH 4180 - Complex Analysis
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Students interested in the applications of complex analysis should consider MATH 4220 rather than MATH 4180; however, undergraduates who plan to attend graduate school in mathematics should take MATH 4180.
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Theoretical and rigorous introduction to complex variable theory. Topics include complex numbers, differential and integral calculus for functions of a complex variable including Cauchy's theorem and the calculus of residues, elements of conformal mapping."
"MATH 4220 - Applied Complex Analysis
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Undergraduates who plan to attend graduate school in mathematics should take MATH 4180 instead of 4220.
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Covers complex variables, Fourier transforms, Laplace transforms and applications to partial differential equations. Additional topics may include an introduction to generalized functions."
A good course to do after PDE is one that involves applied functional analysis, in which you can apply the theory of Banach and Hilbert spaces to problems involving ODE, PDE and distributions.
