Real semialgebraic sets are sets definable in the language of the reals: $(\mathbb{R},0,1,+,\cdot)$, which has as a definitional extension $(\mathbb{R},0,1,+,\cdot,\leq)$ by the useful fact that $a< b$ iff there exists $t\in\mathbb{R}$ such that $(b-a)t^2=1$.
For better or worse, the same trick does not work for the language of the complex numbers. Is there a name for the sorts of sets and functions definable over the complex numbers equipped with the absolute value symbol? I'm imagining something like definable sets over local fields, where the logical structure provides you a valuation symbol and an ordering on the valuation group. Have the complex version of these been studied at all?
If you add $|\cdot|$ to the language of $\Bbb{C}$ and interpret it as absolute value, then the real line $\Bbb{R}$ becomes definable as $\{z \mid |z| = z \lor |z| = -z\}$ and you are looking at $\Bbb{C}$ as an $\Bbb{R}$-algebra. The definable sets in $\Bbb{C}^n$ then coincide with the semialgebraic subsets of $\Bbb{R}^{2n}$ (as Lord Shark surmised in his comment).