I think I know what you mean by 'what is the complete form, of Je m'appelle?
If the verb (appeller) did not begin with a vowel, then the sentence would be je me + verb. For example - je me lave = I wash (myself).
But, the French avoid having two vowels next to each other in this situation so it has to be je m'appelle not 'je me appelle,' which is wrong because the 'e' of 'me,' would run into the 'a' of the verb appeller.