The structure of continuous and discrete-time linear behaviors over quaternions, i.e., the solutions of linear difference and differential equations with quaternion coefficients, was investigated in a previous paper. However, in a particular case, which is peculiar to quaternions, a basis of the vector space of solutions was just outlined and the rather complex scheme for its construction was only partially described by examples. Using a different approach, an equivalent basis will be proposed in this contribution together with an algorithmic construction.