We explore the existence and asymptotic stability of equilibrium point for a class of quaternion-valued BAM neural networks with time-varying delays. Firstly, by employing Homeomorphism theorem and a contradictory method with novel analysis skills, a criterion ensuring the existence of equilibrium point of the considered quaternion-valued BAM neural networks is acquired. Secondly, by constructing two Lyapunov functionals, a criterion assuring the global asymptotic stability of equilibrium point for above discussed quaternion-valued BAM is presented. Applying a contradictory method to study the equilibrium point and applying two Lyapunov functionals to study stability of equilibrium point are completely new methods.