The complex autoregressive model we proposed as a shape descriptor has such a merit that the complex autoregressive coefficients are invariant for translation, rotation and dilatation of shapes. Those are also invariant for a start point of boundary tracking. In this paper, several distances for retrieving similar shapes are proposed based on the model. The distances can be seen to be natural extension of the distances based on the conventional autoregressive model. The properties of the distances are discussed, and effectiveness of the distances for shape retrieval is verified by some experiments.