Let $\mathbb{C}\_{an}$ be the expansion of the structure $(\mathbb{C}; +,-,×,0,1)$ by adding the restricted complex analytic functions. This is the complex analog of the familiar $\mathbb{R}\_{an}$ in O-minimality.
What do we know about the model theory of $\mathbb{C}\_{an}$? Model-completeness? Quantifier-elimination? etc.
PS. It seems Rückert's Nullstellensatz gives us model-completeness (This came to my mind just now) Is this true?



Model theory of the restricted complex analytic functions
The structure $\mathbb C\_{an}$ is bi-interpretable with $\mathbb R\_{an}$. To see this, take for example the restriction of the complex exponential function to the unit square S with corners 0,1,i,1+i. Then from this restriction you can define S as the set of points where the restricted function takes a non-zero value. Then the set of complex numbers $x$ satisfying $\forall y[y \in S \to xy \in S]$ is the unit interval [0,1]. Then taking $\pm$ and $1/x$ you can define $\mathbb R$.


So while many people do study restricted holomorphic functions, they do so usually in the context of o-minimality. 


