>>7625342It really depends on the content of the particular course. "Complex variables" can mean anything from a computational, non-proof based survey course for engineers to a graduate-level-something being taught out of Ahlfors.
Kreyszig's functional analysis text, for one, doesn't assume you've had a prior exposure to analysis though it's on the level of rigorous pure math.
You'll find a lot that doesn't have specific prerequisites. More important by far is mathematical maturity. A lot of unis use analysis as a first real encounter with abstract math, and double it up as an intro to proofs course. It's one of the easiest to motivate ("putting high school calculus on a rigorous foundation") but there's no reason it's the only correct choice. I will say that trying to understand the *theory* of differential equations or bona fide complex analysis without your standard advanced calculus/real analysis background is asking for trouble.
