Many computations associated with the two-parameter Cauchy model are shown to be greatly simplified if the parameter space is represented by the complex plane rather than the real plane. With this convention we show that the family is closed under Mö bius transformation of the sample space: there is a similar induced transformation on the parameter space. The chief raison d'être of the paper, however, is that the two-parameter Cauchy model provides an example of a nonunique configuration ancillary in the sense of Fisher (1934), such that the maximum likelihood estimate together with either ancillary is minimal sufficient. Some consequences for Bayesian inference and non-Bayesian conditional inference are explored. In particular, it is shown that conditional coverage probability assessments depend on the choice of ancillary. For moderate deviations, the effect occurs in the Op(n−1) term: for large deviations the relative effect is Op(½)