Is it obligatory to complete treatment with a Christian doctor, or to leave him and go to a Muslim doctor after discovering another Muslim doctor?

If a Christian doctor is trustworthy, reliable, and skilled, and improvement in treatment is achieved through him, then there is no harm in continuing with him. There is no obligation to leave him and search for a Muslim doctor. Muslims, both in the past and present, have sought help from skilled non-Muslim doctors. It is permissible to seek assistance from a disbeliever in medicine and similar fields if he is an expert and trustworthy, just as the Prophet, peace and blessings be upon him, hired Abdullah ibn Urayqit al-Du'ali as a guide during the Hijra while he was a disbeliever.