The Nestorians were a branch of Christianity who followed the doctrine of the 5th century Nestorius, Bishop of Constantinople. Contrary to orthodox belief that Christ is one person with two natures, divine and human, Nestorius asserted that Christ was two person, God the son and Jesus the man.
The Nestorian church rejected icons and images and only used simple crosses at the entrance of churches and tombs. It is unlikely that a Nestorian cross as such ever existed. In his travels, however, Marco Polo famously describes the cross on the tomb of St. Thomas near Madras, India as the Nestorian cross. This appears to be a Cross Bottony, above which appears the Holy Spirit in the form of a dove.