This question stems from a discussion with a friend about the apparent jump in rigidity of smooth geometry to complex geometry. The word rigidity here is somewhat vague; I mean this in the sense it imposes significantly more restrictions on behavior of these objects (as H. R. Miller once said in a lecture, "groups are rigid, but spaces are squishy").
In complex analysis, asking for a complex derivative is an incredibly restrictive condition. Not only are these functions everywhere analytic, but we get results such as Cauchy's theorem, Liouville's theorem, and Hadamard factorization as "basic" consequences. The theory of functions of a complex variable has a very different character to functions of a real variable.
If we're building a manifold, and require the charts to produce holomorphic transition maps, we get a similarly restrictive condition. These manifolds behave more like algebraic varieties over $\mathbb{C}$ than smooth manifolds. Complex manifolds are automatically smooth and canonically oriented. Most spheres fail to be complex manifolds (except $S^2$, the Riemann sphere), hinting that complex structure is a somehow "rare" phenomenon.
Ultimately, I want to know what it is about having derivatives on $\mathbb{C}$ that causes this apparent rigidity. Some random candidates that come to mind are ideas from the proof of Cauchy's theorem, algebraic closure of $\mathbb{C}$, and the fact that $\dim_{\mathbb{R}} \mathbb{C} = 2$. There really aren't any good analogies of this phenomenon that I know of, but I'd be interested in hearing about them if they exist.
As commenters have pointed out, real analytic geometry behaves similarly to complex geometry, though real analytic geometry can be done on a seemingly wider class of manifolds, while complex manifolds are somehow "rare". Presumably this is to do with the fact that holomorphic functions, in addition to being analytic, satisfy an elliptic PDE (the C-R equations). I have little background in PDEs, though, so the fundamental reason for this "rigidity" is still somewhat mysterious to me.
