Complex analysis is the mathematical analysis of complex-valued analytic (typically) functions of a complex variable, of several complex variables, or on a complex analytic manifold.
Since a complex number can be understood as a pair of real numbers, this would naively reduce to analysis of pairs of functions of an even number of real variables; however by complex analysis we mean mathematical analysis which takes into account limits and derivative which do not depend on the real line in a complex plane on which we approach a point. This leads to the notions of holomorphic function, meromorphic function, etc. which are the main subject of complex analysis. However, there are connections to such real-analytic notions as harmonic analysis, and the geometric approach to complex analysis builds on the theory of smooth functions.
Wikipedia (English): Complex analysis, Several complex variables
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