Nonlinear complex principal component analysis (NLCPCA) can be performed by complex-valued neural network models to extract both linear and nonlinear relations between variables in 2-D vector fields such as horizontal winds or currents. The NLCPCA is a generalization of the complex principal component analysis (CPCA) which determines only linear relationships between complex variables. The NLCPCA is a complex-valued neural network algorithm for data dimensionality reduction and feature extraction in the complex domain. An important distinction is drawn between 2-D NLPCA, that is for dimensionality reduction in the real domain to two real variables, and the NLCPCA which reduces the dimension of several complex variables to a single complex variable. A theory of NLCPCA is developed and applied to analyze the tropical Pacific wind field. The linear CPCA of the tropical Pacific wind field describes the El Niño - La Niña  phenomenon in about four modes, whereas the NLCPCA model extracts the full 2-D phenomenon in a single nonlinear mode.