He becomes more confident. As a god, Naruto has powers that far exceed his previous abilities, so he no longer has to doubt himself in battles. For example, he can face the strongest opponents without hesitation.
He may become more detached from his old self. With godhood comes a different way of thinking. He might view mortal problems as trivial at first. But as the story progresses, he realizes that his roots as a normal ninja are important, and he tries to balance his new divine nature with his old human emotions.
Naruto's sense of responsibility expands greatly. Since he has god - like powers, he feels a need to protect not just his village, but the entire ninja world and even other realms if the story goes that way. He might start making decisions for the greater good, which could sometimes put him at odds with his friends who don't understand his new perspective.
His sense of justice could become more absolute. As a god, he may not tolerate any form of evil. He would strive to purge the ninja world of all dark elements. His power would allow him to enforce his version of justice, which could be both a good thing in terms of eliminating real threats, but also a bit scary as it might limit the free will of some ninjas who are not truly evil but just made some mistakes.
His confidence would skyrocket. Once Issei becomes a dragon god, he has new powers and a higher status. This would make him more self - assured. He would also become more responsible as he has to look after things related to his new position.
Well, if Naruto is banished and becomes a monk in fanfiction, it could make him more introspective. He might become calmer and more in tune with his spiritual side. His once boisterous nature could be toned down as he devotes himself to monk - like practices such as meditation and self - reflection. This change could also lead to him having a different perspective on his past relationships, both the good and the bad.
His friends might initially be shocked and a bit afraid. They've known Naruto as their equal, and now he's a god. But eventually, they would come to accept him. Sasuke, for example, might see Naruto's new status as a challenge to get stronger himself.
His confidence might increase even more. Naruto was always confident as a ninja, but as a sorcerer, he has new powers at his disposal. He would feel even more capable of taking on challenges. For example, if he faces a powerful sorcerer enemy, instead of being intimidated, he would be eager to test his new abilities.
He might become more patient. As a medic, he has to be careful and take his time with patients, which is different from his impulsive fighting style.
His relationship with Sasuke would be very different. Sasuke might view Naruto with a new sense of awe and respect. Instead of their rivalry, Sasuke could become Naruto's most loyal follower or partner in maintaining the balance of the new world Naruto as a primordial god would oversee. With his friends like Sakura and Kakashi, Naruto would be more of a protector. They would look up to him not just as a friend but as a divine being.
He would become more cynical. Instead of his usual optimistic self, he'd view the world in a darker light. His trust in people would be severely damaged.
Naruto's normally positive and upbeat character might become more hardened. He'd be more cynical after seeing so much destruction. His trust in others might be shaken, especially if he's been betrayed during the apocalypse.
He becomes cold towards others. He no longer shows the warm smiles and friendly greetings that he used to. Instead, he might ignore people or give them cold stares.