An adaptive complementary sliding-mode control (ACSMC) system with a multi-input-multi-output (MIMO) recurrent Hermite neural network (RHNN) estimator is proposed to control the position of the rotor in the axial direction of a thrust active magnetic bearing (TAMB) system for the tracking of various reference trajectories in this study. First, the operating principles and dynamic model of the TAMB system using a linearized electromagnetic force model is derived. Then, a conventional sliding-mode control (SMC) system is designed for the tracking of various reference trajectories. Moreover, a complementary sliding-mode control (CSMC) system is adopted to reduce the guaranteed ultimate bound of the tracking error by half while using the saturation function as compared with the SMC. Furthermore, since the system parameters and the external disturbance are highly nonlinear and time-varying, the ACSMC is proposed to further improve the control performance and increase the robustness of the TAMB system. In the ACSMC, the MIMO RHNN estimator with estimation laws is proposed to estimate two complicated dynamic functions of the system on-line. In addition, a robust compensator is proposed to confront the minimum approximated errors and achieve the robustness. Finally, some experimental results for the tracking of various reference trajectories show that the control performance of the ACSMC is significantly improved comparing with the SMC and CSMC.