The theory of electrodynamic space-time relativity (TESTR) is the study of the transformation of time and space between two electrodynamic inertial frames of reference, which have both inertial velocity difference and electric potential difference. It is a fundamental space-time theory of theoretic physics based on the Einstein’s special theory of relativity, the electric potential limit postulate and the high-precision experimental facts of the inversion proportional square law of Coulomb’s force. It founded new physical space-time concepts, for example, the electrodynamic space-time is five-dimensional which is composed of quaternion space, and time. It also proposed some new basic concepts of physics, such as electric potential limit, quaternion electric potential and etc., revealing the inherent relationships between electric potential-velocity and time-space. This paper discusses in detail the process of establishing the theory of complex electrodynamic space-time relativity and theory of quaternion electrodynamic space-time relativity as well as their various conversions and transformations. At the same time, the basic effects of TESTR were discussed, obtaining the equations for superposition of quaternion velocity and potential of TESTR. It predicts some important new space-time change effects, which provides theoretical basis for the experimental validation of TESTR. This is the fundamental theory of the author’s series. Keyword: theory of electrodynamic space-time relativity, theory of complex electrodynamic space-time relativity, theory of quaternion electrodynamic space-time relativity, theory of electric potential relativity, special theory of relativity, electric potential limit, complex speed, complex electric potential, quaternion velocity, quaternion electric potential, quaternion, electrodynamic space-time, five-dimensional space-time, the time expansion effect of electric potential, electrodynamic space-time effects