studied the electromagnetic theory with the hyperbolic quaternion. K. Morita [@morita] researched the quaternion field theory. S. M. Grusky [@grusky] etc utilized the quaternion to investigate the time-dependent electromagnetic field. H. T. Anastassiu [@anastassiu] etc applied the quaternion to describe the electromagnetic feature. J. G. Winans [@winans] described the physics quantities with the quaternion. Meanwhile the complex quaternion has certainly piqued scholars’ interest in researching the gravitational theory. J. Edmonds [@edmonds] utilized the quaternion to depict the wave equation and gravitational theory in the curved space-time. F. A. Doria [@doria] adopted the quaternion to research the gravitational theory. A. S. Rawat [@rawat] etc discussed the gravitational field equation with the quaternion treatment. Moreover a few scholars applied the octonion analysis to study the electromagnetic and gravitational theories. M. Gogberashvili [@gogberashvili] discussed the electromagnetic theory with the octonion. V. L. Mironov [@mironov] etc applied the octonion analysis to represent the Maxwell’s equations and relevant physics features. S. Demir [@demir2] etc applied the algebra of octonions to study the gravitational field equations. In the paper, we shall focus on the application of the complex quaternion on the electromagnetic and gravitational theory field.

At present there are mainly three description methods for the 