This work explores an analytic continuation of relativistic kinematics through the transformation v → iv, leading to a modified, non-singular Lorentz factor. The resulting structure is interpreted as a Euclideanized kinematic regime analogous to Wick rotation in quantum field theory. The framework suggests a connection between relativistic kinematics and Euclidean field theory, where imaginary velocity corresponds to non-classical or off-shell configurations. An effective scalar field description is introduced, linking the Euclidean sector to compact field variables and semiclassical gravitational dynamics. Possible implications for cosmology, quantum gravity, and gravitational wave propagation are discussed. This work is a theoretical exploration and has not yet been peer-reviewed.