Instead of thinking of real analysis as a zoo of weird exceptions, it's probably more accurate to think of complex analyic functions as the exceptions. For example, when viewed as a two dimensional mapping from the plane to itself, complex-analytic functions are conformal (angle-preserving) whereas most differentiable mappings from the plane to itself are not.
On the other hand, complex analysis explains things that aren’t obvious in real analysis, like why the radius of convergence for a continuous function on the real line might be 1 (turns out there are singularities off the real line on the complex plane).
