Nestorianism was born from an early schism in the Church
Nestorians are a sect of followers of Nestorius who denied the hypostatic union and maintain the existence of two persons in Christ; the man Jesus and the divine Son of God, rather than as one divine person, unified, with two natures. Hence Nestorianism has been considered heretical by most Western and Orthodox churches.
Nestorius (c. 386–451) was Archbishop of Constantinople and his view of Christ led to the Nestorian schism, separating the Assyrian Church of the East from the Byzantine Church. The difference twixt Christ as two persons or one person is most significant for the Christian's understanding of salvation and the theology of the Eucharist. Nestorians rejected the concept that it was God who was crucified; rather it was the humanity of Christ who suffered.
Nestorians were active missionaries and in the 7th and 8th centuries spread Christianity eastwards to Persia, Central Asia and East Asia, including China, Mongolia, Korea and Japan. Their influence waned, or the church was crushed, in the Far Eastern realms but remained relatively strong in the rest of Asia.
As for the so-called Nestorian Cross, variations of this design have been found on Nestorian or Assyrian tombs in Asia and follows the lines and interpretation of the Budded Cross.