Principal component analysis (PCA) has been generalized to complex principal component analysis (CPCA), which has been widely applied to complex-valued data, 2-dimensional vector fields, and complexified real data through the Hilbert transform. Nonlinear PCA (NLPCA) can also be performed using auto-associative feed-forward neural network (NN) models, which allows the extraction of nonlinear features in the data set. This paper introduces a nonlinear complex PCA (NLCPCA) method, which allows nonlinear feature extraction and dimension reduction in complex-valued data sets. The NLCPCA uses the architecture of the NLPCA network, but with complex variables (including complex weight and bias parameters). The application of NLCPCA on test problems confirms its ability to extract nonlinear features missed by the CPCA. For similar number of model parameters, the NLCPCA captures more variance of a data set than the alternative real approach (i.e. replacing each complex variable by 2 real variables and applying NLPCA). The NLCPCA is also used to perform nonlinear Hilbert PCA (NLHPCA) on complexified real data. The NLCPCA is applied to the tropical Pacific wind field and to the sea surface temperature field, thereby extracting the El Niņo-Southern Oscillation signal.