Disclaimer: "soft" answer.
While Gordon's comment is very true, I think it's worth adding that it also depends heavily on the fund, its approach, and its strategies. Maybe you find a place that does some signal processing stuff, in which case complex definitely helps. But there are a lot of places where you could get by without being well-versed in real/complex analysis as long as you're intellectually disciplined enough to understand and think critically about any assumptions underlying the stuff you're doing, which will likely mean learning some math at some point (be it reviewing lecture notes or reading a journal article).
Experience with mathematical analysis will definitely help with that, but it's probably worth keeping in perspective for the simple reason that you can't prove a trading strategy will work as planned in the same way you can prove Euler's Identity. It's a matter of doing the proper research, which could well entail a good deal more probability/statistics than math. You could easily understand all the math under the hood and still fail to stress-test a model with out-of-sample data or with relaxed assumptions and end up losing a lot of money. Hence why I feel it's worth at least mentioning that ultimately the ability to think critically matters about as much as what you're thinking about.
TLDR: Intellectual discipline matters, regardless of subject matter.