What is the most useful to learn out of complex analysis and differential equations for undergraduate studies in physics? Next year I'm planning to start on my bachelor's in physics, however, I have already started taking some undergraduate courses in mathematics and next semester I will have to choose between complex analysis and differential equations - I will take the other course at some other time, but that will at least be a year away. 
I know both subjects are important in physics, but my question is which subject will have the largest impact when I start on my bachelor's degree. 
 A: Complex analysis is unlikely to be of any use at all in undergraduate physics classes. Differential equations are used constantly. All of physics is expressed in terms of differential equations, such as Maxwell's equations and the Schrodinger equation.
A: As an undergraduate, having a decent understanding of complex numbers, rather than complex analysis, goes far. Complex analysis - complex functions and integrals of complex functions in the complex plane and power series of complex functions and the multiple of theorems, etc... - is not very useful in an undergraduate curriculum, unless it is an advanced undergrad curriculum. 
However, what's more useful for an undergrad is a good understanding of the complex plane and how to use it to make problems easier. The main application is in quantum mechanics, where we are dealing with complex numbers and functions ubiquitously.  Most of the relevant information necessary for introductory quantum mechanics regarding complex numbers is usually covered by an intro. text book, and I strongly recommend David McynTyre's book, and/or Griffith's book. For more mathematical treatments of complex numbers, see this post.
