I'm majoring in math and am considering graduate school in either pure math, applied math, or stats. The problem is that, due to schedule constraints, I can only take either Complex Variables or Topology. How necessary is Topology for pure math admissions? Is complex Vars. more necessary? Does either matter for stats and applied? Thanks.
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Complex variables is useful in both pure and applied mathematics, to a roughly equal extent. With a few significant exceptions (e.g. persistent homology) topology is used very little in applied mathematics.
For instance, in the graduate program in mathematics at the University of Georgia, most students concentrating in pure mathematics take a qualifying exam in topology, and most students concentrating in applied mathematics take numerical analysis or probability instead. All students are currently required to take a qualifying exam in complex analysis.
I have much less experience with statistics, but I am almost certain that complex analysis is more useful there than topology. (What I don't know is much complex analysis is actually used!)
In my experience (I am the chair of graduate admissions of the mathematics department at UGA) complex analysis and topology look equally good on transcripts. (Both would look better, of course -- it sounds like you knew that already.)
So if it's really a tossup, taking complex analysis holds more doors open for you. However, both of these are courses that beginning graduate students (at least in the US) quite often take, so it's not exactly a life-changing decision. In fact, I would advise you to take whichever one sounds more interesting to you. Good luck.
I doubt that it matters. It is good, however to have a well rounded undergraduate education. If you already have a lot of Analysis, maybe Topology would be marginally better. But the difference between Real and Complex Analysis is quite profound - especially at the insight level. Likely you will find graduate courses available in whichever one you don't take now.