Kobi Peterzil and Sergei Starchenko have developed a  theory of o-minimal complex analysis.   Just as with  the classical approach to complex analysis, one may interpret  the complex numbers as the set of ordered pairs of real numbers  with addition and multiplication defined by the usual rules  involving their real and imaginary parts.  Of their results in this  area, their algebraicity theorem, which asserts that if a subset of  Cn is complex analytic (meaning that it is closed and is locally defined by the vanishing of finitely many  complex analytic functions) and definable in some o-minimal expansion of the real field, then it must be algebraic, that is defined by the  vanishing of polynomial equations, is the most striking result and  has been strong consequences in the study of functional transcendence and homogeneous dynamics. 