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
John B. Conway, Functions of one complex variable, Springer 1978; Functions of one complex variable II, GTM 159
Lars V. Ahlfors, Complex analysis, McGraw-Hill, 1966.
Raghavan Narasimhan, Complex analysis in one variable, Birkhäuser, 1985.
Lars Hörmander, An introduction to complex analysis in several variables, North-Holland
Steven G. Krantz, Geometric function theory: explorations in complex analysis, Birkhäuser
Robert C. Gunning, Hugo Rossi, Analytic functions of several complex variables, AMS Chelsea Publishing
Douglas N. Arnold, Complex analysis, lecture notes, pdf
Elias M. Stein?, Rami Shakarchi, Complex analysis, Princeton University Press 2003, 2012