In this paper, kurtosis-based complex-valued real-time recurrent learning (KCRTRL) and kurtosis-based augmented CRTRL (KACRTRL) algorithms are proposed for training fully connected recurrent neural networks (FCRNNs) in the complex domain. These algorithms are designed by minimizing the cost functions based on the kurtosis of a complex-valued error signal. The KCRTRL algorithm exploits the circularity properties of the complex-valued signals, and this algorithm not only provides a faster convergence rate but also results in a lower steady-state error. However, the KCRTRL algorithm is suboptimal in the processing of noncircular (NC) complex-valued signals. On the other hand, the KACRTRL algorithm contains a complete second-order information due to the augmented statistics, thus considerably improves the performance of the FCRNN in the processing of NC complex-valued signals. Simulation results on the one-step-ahead prediction problems show that the proposed KCRTRL algorithm significantly enhances the performance for only circular complex-valued signals, whereas the proposed KACRTRL algorithm provides more superior performance than existing algorithms for NC complex-valued signals in terms of the convergence rate and the steady-state error.