We introduce new families of four-dimensional Ricci solitons of cohomogeneity two with collapsing ends. In a local presentation of the metric conformal to a product, we reduce the soliton equation to a degenerate Monge-Ampère equation for the conformal factor coupled with ODEs. We exhibit explicit solutions and obtain abstract existence results for complete expanding solitons and singular shrinking and steady ones. These families of Ricci solitons specialize to classical examples of Einstein and soliton metrics.