An increasing number of scholars have proposed many schemes to mitigate the Kerr nonlinearity effect restricting the transmission capacity of optical fibres. In this paper, we proposed a complex principal component analysis-based complex-valued fully connected neural network (P-CFNN) model aided by perturbation theory and demonstrated it experimentally on a dual-polarization 64-quadrature-amplitude modulation coherent optical communication system. What we believe to be a novel complex principal component analysis (CPCA) algorithm applied to complex-valued fully connected neural network (CFNN) is designed to further reduce the computational complexity of the model. Meanwhile, an equivalent real-valued fully connected neural network (RFNN) with the same time complexity as a CFNN is proposed for fair performance comparison. Under all launched optical powers, the performance of the P-CFNN equalizer is the best among all comparison algorithms, and the maximum ΔQ-factor compared to without employing the nonlinear compensation algorithm reaches 3.94 dB. In addition, under the constraint of the same Q-factor, we confirmed that the proposed P-CFNN obtained a 40% reduction in time complexity and a 70% reduction in space complexity compared with the PCA-based RFNN, which also proved the very large application prospect of the P-CFNN equalizer in optical fibre communication systems.