What is a Christian Buddhist?

Answer 1

Both Christian and Buddhist scholars would see that the differences in their positions preclude dual membership.

For a Christian there is a God with all sorts of control and mystic powers that runs the world and their lives (a position held by all the Abrahamic faiths). Particularly the Christians believe in Jesus an avatar of their God. The ingoing position is that everyone is evil from the get go (original sin" and has to be "saved" and that those who aren't saved go to hell forever and the good folks to to heaven . Christians only get one chance to "do it right."

To a Buddhist everyone is born with a Buddha nature (basically a goodness) which they themselves can cultivate with no godly assistance until they learn to run their lives in manner that is satisfying to themselves and the world in general. After this enlightenment they may or may not go to a condition of Nirvana (this is their choice). They get many chances (reincarnation) to learn how to improve their lives.

With this degree of divergence once could find many difficulties in accepting both positions. Fortunately the Eightfold Path (a Buddhist guideline for proper action) is useful for anyone who want to be a better person and Christians may adopt these positions without being Buddhists.

Answer 2

There cannot be a Christian Buddhist. Christians believe in Jesus Christ, whereas Buddhists believe in different deities.