Mark 8
Judaism in those times was an Eastern mystical religion and reincarnation was a common belief. The Jews of the Diaspora who were more Hellenized, dropped this belief. It could be that it is still held in certain kinds of Jewish mysticism.
Those verses refer to resurrections, not reincarnations.
The concept of resurrection, already glimpsed at in Job 19:25-26, was already implicit in many of the Psalms (take Psalms 16 and 73, for example), was taught explicitly by the 6th century BC in the Book of Daniel, and was widely believed by the time of the Maccabees (circa 167 BC). The Pharisees officially embraced this doctrine, while the Sadducees denied it.
Elijah was taken up to heaven in the chariot of fire (2 Kings), and it was taught by Malachi, last of the Prophets (circa 450-400 BC), that he would return before the coming of the Lord - again, not a reincarnation, but a “descent” from Heaven.
In the case of St. John the Baptist, even Herod, who had him executed, thought that he was “risen from the dead” (i.e. resurrected) and working miracles when he heard about Jesus.
There are apocryphal works such as the “Assumption of Moses” which speak of Moses being taken up into Heaven as well, so a “return of the Prophet” would again have referred to resurrection or descent from heaven, not to reincarnation.
And the Epistle to the Hebrews - written by first-generation Jewish converts to Christianity (modern scholarship says St. Paul plus one of his associates) - clearly says that “it is given for men to die once, and after that the judgement.” (Hebrews 9: 27)
Reincarnation was always a feature in other Eastern religions, such as Hinduism. Classical (pre-70 BC) Judaism did not teach it, neither do Orthodox Jews today embrace it. It has infiltrated “liberal” Jewish and Christian thought through the toxic agency of the “New Age Movement”, but it was never a part of the original Judaeo-Christian tradition.
