What axiom system for the complex numbers is categorical?

In full second-order logic you can characterize $\mathbb{C}$ in the language $(+,\cdot,0,1)$ up to isomorphism using the following axioms:

1. First-order axioms stating that $(M,+,\cdot,0,1)$ forms an algebraically closed field of characteristic zero.
2. A second-order axiom stating that there is a subset $R$, a function $f$ and a relation $<$ such that $(R,+,\cdot,0,1,<)$ forms a Dedekind-complete ordered field and $f$ is a bijection between $R$ and the whole structure $M$.

This theory is categorical. Why? As Olivier Roche alluded to in his answer, the theory of algebraically closed fields of characteristic zero has a unique model in each cardinality $\lambda > \aleph_0$. Moreover, every Dedekind-complete ordered field has the cardinality of $|\mathbb{R}|$, so the models of the theory above are precisely the algebraically closed fields of characteristic zero of cardinality $|\mathbb{R}|$, so they are all isomorphic to $(\mathbb{C},+,\cdot,0,1)$.