An algebraically closed field \(K\) can be conceived as \(R( \sqrt{-1})\), with real closed \(R\). And, \(R\) is orderable by the Artin-Schreier theorem, and so o-minimality is relevant to \(R\), \(K\), and their expansions. In a long first part of this paper, the authors develop a differential part of complex analysis on \(K\). Classical tools, like convergence and integrals are not available, and they resort to topological tools like winding numbers. (The main reference is G. T. Whyburn’s book: Topological analysis. Revised ed. Princeton, N. J.: Princeton Univ. Press (1964; Zbl 0186.55901), the authors state.) Their development is in terms of definable sets and functions, and includes (analogues of) Laurent expansions, residues, the maximum principle, and Liouville’s and Rouché’s theorems. These works are used to obtain a model-theoretic result, as follows. Let \({\mathcal R}\) be an o-minimal expansion of \(R\), and let \({\mathcal K}\) be the expansion of \(K\) defined in \({\mathcal R}\). If \({\mathcal K}\) defines more sets than \(\langle K,+,\cdot\rangle\) does, then \({\mathcal K}\) defines \(R\), too. From this follows a classical result of Chow: An analytic set in \(\mathbb{P}_n (\mathbb{C})\) is algebraic. (For the usual proof, refer, e.g., to P. Griffiths and J. Harris, Principles of algebraic geometry, New York etc.: Wiley (1978; Zbl 0408.14001)).