In this paper we present a Complex-Valued Recurrent Neural Network (CVRNN), trained with a recursive Levenberg-Marquardt (LM) learning algorithm in the complex domain, applying it to the problem of dynamic system identification, and to an adaptive neural control scheme of a nonlinear oscillatory plant. This methodology is applied to two different CVRNN topologies with different kinds of activation functions. Furthermore, we applied the CVRNN identification and control for a particular case of a nonlinear, oscillatory mechanical plant to validate the performance of the adaptive neural controller using the LM algorithm developed throughout this work, compared to a complex-valued Backpropagation learning algorithm. The obtained comparative simulation results using a flexible robot arm gives a good performance of the derived control schemes. The results show some priority of the recursive LM learning over the BP learning, and the use of constructed activation functions in the neural network topology.