Interpreting the Electric Field with Complexified Quaternions

This paper revisits classical electromagnetism using complexified quaternions (biquaternions), an associative algebra isomorphic to $M_2\left(？\right)$ . Building on Maxwell’s original quaternionic insights, we define a generalized four-gradient ${？}_r$ and four-potential $A$ in biquaternionic form. The electric field $E$ and magnetic field $B$ emerge from the anticommutator and commutator of ${？}_r$ and $A$ , respectively. Explicit computation yields the standard expressions for $B=？\times A$ and $E=？？？？\left(1/c\right){？}_{t}A$ plus a gauge-dependent scalar term $\left(1/c\right){？}_t？+？？A$ . In a specific gauge where $A=？？S$ and $？=\left(1/c\right){？}_{t}S$ , this reduces to the d’Alembertian wave equation $$