Doctor, also known as Dr. Lugae, represents Dr. Lugae, a storyline boss in Final Fantasy IV. He and Rubicante destroyed Eblan with an army of monsters, after which Lugae turned the King and Queen of Eblan into monsters.
The Doctor is a relatively easy boss for a high-level party. Barnabas is fought at the same time, and he will never attack the party. He occasionally restores major HP to Barnabas at regular intervals.
It is best to try and defeat him and Barnabas at the same time. If left alone long enough after Barnabas is defeated, Doctor will take control of Barnabas and become Barnabas-Z.