A novel complex echo state network (ESN), utilizing full second-order statistical information in the complex domain, is introduced. This is achieved through the use of the so-called augmented complex statistics, thus making complex ESNs suitable for processing the generality of complex-valued signals, both second-order circular (proper) and noncircular (improper). Next, in order to deal with nonstationary processes with large nonlinear dynamics, a nonlinear readout layer is introduced and is further equipped with an adaptive amplitude of the nonlinearity. This combination of augmented complex statistics and enhanced adaptivity within ESNs also facilitates the processing of bivariate signals with strong component correlations. Simulations in the prediction setting on both circular and noncircular synthetic benchmark processes and real-world noncircular and nonstationary wind signals support the analysis.