witch doctor vs magician

witch doctor and magician both are nouns.

Word	Noun	Adjective	Verb	Adverb
witch doctor	Yes	No	No	No
magician	Yes	No	No	No

As nouns, magician is a hypernym of witch doctor; that is, magician is a word with a broader meaning than witch doctor:

Other hypernyms of witch doctor include necromancer, sorcerer, thaumaturge, thaumaturgist, wizard.

witch doctor (noun)	magician (noun)
someone who is believed to heal through magical powers	one who practices magic or sorcery
	someone who performs magic tricks to amuse an audience