In the paper, the quaternion-valued neural networks (QVNNs) with non-differentiable time-varying delays are considered. Firstly, by using the method of plural decomposition, we decompose the QVNNs into two complex-valued neural networks. Some sufficient criteria in linear matrix inequality (LMI) form are derived to guarantee the existence and uniqueness of the equilibrium point for considered QVNNs by using the homeomorphism mapping principle of complex domain. Secondly, based on applying the free weighting matrix method and constructing appropriate Lyapunov–Krasovskii functional, several conditions are established in LMIs to ensure the the global μ−stability of QVNNs. Finally, by employing the predictor-corrector approach, two numerical examples are provided to show the feasibility and availability of the obtained result.