Equations in the Complex Quaternion Spaces
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\[sec:level1\]INTRODUCTION
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J. C. Maxwell described the physical feature of electromagnetic field with the vector as well as the quaternion. In 1843 W. R. Hamilton invented the quaternion, and J. T. Graves invented the octonion. Two years later, A. Cayley reinvented the octonion independently in 1845. During two decades from that time on, the scientists and engineers separated the quaternion into the scalar part and vector part, to facilitate its application in the engineering. In 1873 Maxwell mingled naturally the quaternion analysis and vector terminology to depict the electromagnetic feature in his works. Recently some scholars begin to study the physics feature of gravitational field with the algebra of quaternions.

The ordered couple of quaternions compose the octonion. On the contrary, the octonion can be separated into two parts. This case is similar to the complex number, which can be separated into the real number and the imaginary number. The octonion is able to be divided into two parts as well, the quaternion and the $S$-quaternion (short for the second quaternion), and their coordinates are able to be complex numbers. For the convenience of description, the quaternion described in the following context sometimes 