Summary: We developed the consistent relativistic biquaternionic formalism for Bateman’s general solutions of the free Maxwell equations which described, in particular, the linked and knotted electric and magnetic Faraday lines of the light beams in terms of two potentials. The algorithm of constructing the corresponding 4-potential, which does not require the solution of the Cabibbo and Ferrari zero field conditions, was found. This opens up possibilities for a canonical quantization of topologically non-trivial light fields and for correct formulation of the problem of a charged quantum particle motion on their background.