Is there more to complex analysis than a two-dimensional potential theory?

Wikipedia entry on Potential theory in two dimensions says the following

From the fact that the group of conformal transforms is infinite-dimensional in two dimensions and finite-dimensional for more than two dimensions, one can surmise that potential theory in two dimensions is different from potential theory in other dimensions. This is correct and, in fact, when one realizes that any two-dimensional harmonic function is the real part of a complex analytic function, one sees that the subject of two-dimensional potential theory is substantially the same as that of complex analysis. For this reason, when speaking of potential theory, one focuses attention on theorems which hold in three or more dimensions. In this connection, a surprising fact is that many results and concepts originally discovered in complex analysis (such as Schwarz's theorem, Morera's theorem, the Weierstrass-Casorati theorem, Laurent series, and the classification of singularities as removable, poles and essential singularities) generalize to results on harmonic functions in any dimension. By considering which theorems of complex analysis are special cases of theorems of potential theory in any dimension, one can obtain a feel for exactly what is special about complex analysis in two dimensions and what is simply the two-dimensional instance of more general results.

My question is, therefore: what exactly is special about complex analysis in two dimensions and what is simply the two-dimensional instance of more general results?

As hinted in the text you quote, a key difference lies in the fact that there are lots of conformal maps in 1 complex variable, and a lot fewer in higher dimension.

Another difference would be the existence in higher dimension of Fatou-Bieberbach domains, proving the Montel's theorem cannot have a strong generalization to higher dimension.

Non exhaustive list of special features of complex analysis in 2 dimensions:

injective holomorphic functions are conformal (preserve angles)
Riemann's uniformization theorem
the measurable Riemann mapping theorem
the $\lambda$-lemma on holomorphic motions
Koebe's distortion theorem
Montel's theorem on normal families
(more geometric flavor, but still): almost everything is a hyperbolic Riemann surface (except for the Riemann sphere, $\mathbb C$, the cylinder $\mathbb C/\mathbb Z$ and complex tori); in particular, every connected open subset of $\mathbb C$ whose complement contains at least 2 points is hyperbolic

Non exhaustive list of instances of theorems holding in higher dimension, besides those you mention: