E. G. Lewis tells us this much in his blog Sowing the Seeds about the early church:
Pilate remained in office for several years after Jesus was executed. He was removed from office in AD 36 by the Syrian governor, Vitellius, for ordering troops into Samaria to attack a peaceful assembly of Samaritans. Pilate left Judea heading for Rome and an official review of his conduct before Caesar. However, by the time he arrived in Rome Tiberius was dead and Caligula had taken his place. What happened next, no one knows.
Some old traditions say Pilate committed suicide in Rome. The Christian Historian, Eusebius, tells the story and attributes it to the former governor’s remorse for the execution of Jesus. One must temper these reports of suicide with the understanding that to the Romans suicide was considered an honorable death. In fact, if someone displeased the Emperor he could order them to kill themselves and, in most cases, they complied. Other sources say he was exiled to Gaul and committed suicide in Vienne and his body was thrown into the Rhône River. There is even a monument at Vienne, called Pilate’s tomb. In Switzerland, near Lucerne, there is a Mount Pilatus. An old tradition states Pilate was banished to the mountain as punishment.
Pilate appears in apocryphal writings such as The Gospel of Peter, The Acts of Pilate, and The Acts of Peter and Paul. They address the question of what happened to Pilate and, in some cases, also depict him converting to the Christian faith. In addition to Saint Claudia, the Coptic Church also lists Pontius Pilate as a saint. How much of this is a pious attempt to show that no one is beyond redemption and how much is fact, is anyone’s guess.