When I studied complex analysis, I could never understand how once-differentiable complex functions could be possibly be infinitely differentiable. After all, this doesn't hold for functions from $\mathbb R ^2$ to $\mathbb R ^2$. Can anyone explain what is different about complex numbers?
As Akhil mentions, the keyword is elliptic regularity.  Since I don't know anything about this, let me just say some low-level things and maybe they'll make sense to you.

A differentiable function $f : \mathbb{R} \to \mathbb{R}$ can be thought of as a function which behaves locally like a linear function $f(x) = ax + b$.  So, very roughly, it is a collection of tiny vectors which fit together.  These tiny vectors can, however, fit together in a very erratic manner.  That's because since you only have to fit one vector to the two vectors that are its neighbors, there is a lot of room for bad behavior.

A differentiable function $f : \mathbb{C} \to \mathbb{C}$ has to satisfy a much more stringent requirement: locally, it has to behave like a linear function $f(z) = az + b$ where $z, a, b$ are complex, which is a rotation (and scale, and translation).  So, very roughly, it is a collection of tiny rotations which fit together.  Now one rotation has a continuum of neighbors to worry about, and it becomes much harder for erratic behavior to persist.
