In this talk, we discuss the existence, uniqueness, and gradient regularity properties of the complex valued elliptic p-Laplace equations. The framework for the linear equation was studied by Cohen, DeVore, and Schwab in [Acta Numer, 2015], [Found. Comput. Math, 2010], [Anal. Appl, 2011], where they extended real-valued linear equations to the complex setting to obtain numerical results for parametric PDEs.