In the field of artificial neural networks, the Kolmogorov–Arnold Network (KAN), which is inspired by the Kolmogorov–Arnold representation Theorem (KAT), has demonstrated outstanding performance in function fitting. Complex-Valued Kolmogorov–Arnold Network (CVKAN) transfers KAN into the complex domain, providing superior capabilities in complex-valued function fitting. In this paper, we formulate a complex-valued KAT that provides theoretical support for this transformation. Also, given the high suitability of modulus-based activation functions in complex-valued Neural Networks, we propose a ModELU-based CVKAN, which replaces the \(\mathbb {C}\) SiLU residual function in CVKAN with the ModELU function. Experiments demonstrate that our method outperforms CVKAN in function fitting in terms of accuracy and stability. Furthermore, we adopt RBFs with learnable shape parameters in ModELU-based CVKAN, replacing the previous fixed ones. This replacement enhances the model’s performance in function fitting.