Yes, Purgatory is a real, albeit tiny, unincorporated community in Baca County, southeastern Colorado.
The river was named Río de las Ánimas en Purgatorio ("River of Souls in Purgatory") by Spanish explorers—likely referencing a tragedy or the harsh landscape. The settlement adopted the name.
Purgatory is essentially a ghost town / rural hamlet. A handful of homes, a church, and the Purgatory School (historic) remain. No commercial services. It sits along County Road KK. The surrounding area is ranching country (cattle, wheat).
Other "Purgatory" place names in the U.S.: Purgatory Creek (CO), Purgatory Mountain (NC), Purgatory Falls (NH)—but this is the only populated place named simply "Purgatory."