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A complex-valued gradient based algorithm for training nonlinear complex-valued finite impulse response (FIR) filters is derived. The proposed complex error-adaptive normalised nonlinear gradient descent (CEANNGD) and smoothed CEANNGD (SCEANNGD) algorithms are an improvement on the complex nonlinear gradient descent (CNGD) and the complex normalised nonlinear gradient descent (CNNGD) algorithm by including an adaptive term in the normalised learning rate of the CNNGD. This is achieved by performing a minimisation of the complex-valued instantaneous output error that has been approximated via a Taylor series expansion, which makes it suitable for the processing of nonlinear and nonstationary signals. Experiments on complex-valued coloured and nonlinear signals show that the CEANNGD and SCEANNGD algorithms outperform the standard CNNGD and CNGD algorithms.