I've always thought that the English I was much more related to the Norse Jag. I really can't see the change ic > I (mainly considering that ic should be spelled /itch/, I guess).
The Scandinavian
jag and
jeg are descended from the same original proto-Germanic word as German
ich. Anglo-Saxon was a West Germanic language, so Modern English is much more closely related to German, Dutch, Frisian and Yiddish than it is to the North Germanic (Scandinavian) languages.
The word in the more-or-less single Old German language of 2,000 years ago--or the German dialect spectrum across northwestern Europe--was something like
ik, which is preserved in Dutch and Frisian. It was usually spelled
ic in Anglo-Saxon and early Middle English, although it was also sometimes
ik and I've read one article that said it was occasionally spelled
ich. But it was pronounced with a K, not the German fricative CH or the modern English affricate CH sound, so there would be no reason to spell it "itch."
When it lost the final consonant, the short I turned into a long I, which at that time was the cardinal vowel I as in Latin. Then it got caught in the massive (and rather bizarre if you ask me) vowel shift that was one of the hallmarks of the transition from Middle English to Modern English, and it became what we now call a long I in English, the sound that most languages transcribe as AI.
Because many Norsemen came to England during the Middle Ages--for trade, plunder, occupation and peaceful immigration--our language absorbed quite a few words from Old Norse. But "I" is not one of them. Its origin is authentic Old German.
The first-person singular pronoun is a remarkably stable artifact from the original Indo-European language ca. 4000BCE. Latin and Greek
ego and Russian
ya are the same word. The objective case, starting with M, as in English "me" and Russian
mnye, is nearly universal in the entire Indo-European family. In Hindi and Irish it's used in the nominative.