In this paper, by starting from basic quaternion algebra properties and algorithms, we develop a comprehensive set of properties to ensure the uncertain quaternion-valued neural networks can receive synchronization and quasi-synchronization goals. By endowing the classic Lyapunov technique, several sufficient criteria for the synchronization and quasi-synchronization analysis of the addressed model are proposed by means of two simple and rigorous control strategies. Particularly, lexicographical ordering approach is proposed in this paper, which can be employed to determine the “magnitude” of two different quaternion-valued. Finally, we have numerical evidences that the mathematical model and the conclusions presented are validate.