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 theoretical physics based on the Einstein’s special theory of relativity (STR), the electric potential limit postulate and the high-precision experimental facts of the inversion proportional square law of Coulomb’s force. It also proposed new basic physical concepts, such as electric potential limit, quaternion velocity, quaternion electric potential and etc., revealing the inherent relationships between the quaternion velocity, the quaternion electric potential 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, proved that the special theory of relativity is a special case of TESTR, and found another important special case: the theory of electric potential relativity (TEPR). The basic effects of TESTR were also discussed, proposing the equations for superposition of quaternion velocity, and quaternion potential of TESTR respectively. It predicts some important new space-time change effects, which provides theoretical basis for the experimental validation of TESTR.