We present the results of a systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field. Including the well-known example y 2 = x 5 − 1 y^2 = x^5 - 1 we find 19 non-isomorphic such curves. We believe that these are the only such curves.