$\mathbb C$ is an algebraic closed field with characteristic $0$, hence $Th(\mathbb C)$ is a recursive satisfiable complete theory, thus recursive axiomatizable. So if $\mathbb N$ is definable in $\mathbb C$, then $+$,$\cdot$ are definable, hence $0$,$1$,$<$,$S$(successor),$E$(power) are definable too.
The theory of algebraically closed fields of characteristics zero is strongly minimal.
This means that every definable set [in every model] is finite or co-finite.
I took this fact from: A Guide to Classical and Modern Model Theory, p.77.
If you search for a reference about minimal and strongly minimal theories I am sure you can find more about this, including various proofs if you are interested in them.