What are two stories that were only recorded in Luke's Gospel?

The story of the birth of John the Baptist is unique to Luke. Only Luke has John the cousin of Jesus.

Luke's nativity story is unique to Luke's Gospel. Only Luke has the angel appear to Mary to tell her of that she would be the mother of Jesus. Only Luke has Joseph and Mary travel from Nazareth to Bethlehem for a census. Only Lukehas the poor shepherds come to worship the baby Jesus. And only Luke has the young family return peacefully to Jerusalem and Nazareth shortly after the birth of Jesus.

The story of the young Jesus talking to the leaders in the temple is unique to Luke in the gospel record, but closely parallels a similar story in the autobiography of Josephus.

The story of Jesus ascending bodily to heaven on the evening following his resurrection is unique among the gospels to Luke. There is however a very similar account in Acts of the Apostles, written by the same author, of Jesus ascending bodily to heaven after forty days.

The story of Jesus seeing the disciples fishing but catching nothing, and of him telling them to cast the net on the other side, is often thought to be unique to Luke, but is also in John's Gospel. John moves it to the end of the Gospel, as the last resurrection appearance.