The objective of this paper is to present the complex analysis method for the free vibration of general anisotropic rotors. The approach developed in this work represents the natural mode of the rotor as the sum of two sub-modes, one that rotates in the forward direction and the other in the backward direction. It is shown that the natural mode has to be described as such to satisfy the complex equation of motion of general rotor systems. Physical interpretation of results from the analysis of a model of the anti-symmetric motion of the rigid rotor shows that the complex mode contains modal directivity information as well as the conventional modal information. Proposed representation of the natural mode enables one to make clear definition of the forward mode and the backward mode, and more importantly enables one to complete the complex rotor analysis procedure.