Abstract
Within the framework of exterior algebra, the concept of time-like quaternions has been previously established. This paper advances beyond the existing structure by elucidating the procedure for constructing time-like quaternions with the feature of complex scalar terms. We delineate on the crucial role of these extended definition of quaternions in formulating Maxwell equations, having properly defined a pure quaternion containing the components of the classical electric and magnetic fields. Through a formal introduction, we describe the approach followed to acquiring time-like quaternions, characterized by having complex scalar term, and their significant relationship with the derivation of Maxwell equations. This topic not only underscores the mathematical intricacies of quaternionic algebra, but also highlights its profound implications in the description of fundamental electromagnetic phenomena.