Using the well known complex representation of the proper Lorentz group $\mathrm{SO}_+(3,1)\cong \mathrm{PSL}(2,\mathbb{C})\cong \mathrm{SO}(3,\mathbb{C})$ we study some Coriolis type effects in Special Relativity and Electrodynamics in close analogy with the more traditional kinematical treatment of the group of spatial rotations. Namely, we begin with the Clifford group of $\mathbb{R}^3$ viewed as a complexification of $\mathbb{H}^\times$ and consider the associated Maurer-Cartan form which yields a complex-valued analogue of the angular velocity characterizing the action of $\mathrm{SO}(3)$ in rigid body kinematics. It appears in a linear ODE written in bi-quaternion form or a Ricatti equation if one works with its projective version instead (or the so-called Rodrigues' vector), either way providing a far richer structure compared to the real case without imposing serious technical obstructions at least in the decomposable setting, which is our main emphasis due to its importance in physics. There are several distinct terms in the non-commutative part of the so obtained connection describing well known effects named after Coriolis, Thomas, Hall and Sagnac. We also consider a restriction to the so-called Wigner little groups $\mathrm{SO}(3)$, $\mathrm{SO}_+(2,1)$ and $\mathrm{E}(2)$ discussing dynamical properties of the electromagnetic field. Some nice constructions, such as geometric phases, Hopf fibrations or the Fubini-Study form, appear naturally in this approach. We also comment on the physical relevance of analyticity and provide specific examples.