In this paper we apply our version of biquaternion math-phys to electrodynamics, especially to moving electromagnetic dipole moments. After a terminological introduction and applying the developed math-phys to the Maxwell environment, we propose to fuse the Larmor angular velocity with the Lorentz Force Law, producing the biquaternion Lorentz-Larmor Law. This might be a more economic way to deal with cyclotron and tokamak related physics, where the Lorentz Force Law and the Larmor angular velocity actively coexist. Then we propose a biquaternion formulation for the energy-torque-hidden-momentum product related to a moving electromagnetic dipole moment. This expression is then used to get a relativistically invariant force equation on moving electromagnetic dipole moments. This equation is then used to get at the force on a hidden electromagnetic dipole moment and for a derivation of the Ahanorov-Casher force on a magnetic dipole moving in an external electric field. As a conclusion we briefly relate our findings to Mansuripurs recent critique of the Lorentz Force Law and the electrodynamic expressions for the energy-momentum tensor.