I think he might discover hidden powers within himself or learn unique jutsu from unexpected mentors. Maybe he'll even get involved in battles to protect the weak and uncover dark secrets along the way.
Well, he could encounter new allies and enemies in different villages and face all kinds of challenges on his own journey.
Perhaps they are leaving together to search for a cure for some new and dangerous disease that has started to affect ninjas. They believe that by traveling the world, they will find rare herbs or meet wise sages who can help them develop a remedy. Naruto's determination and Hinata's knowledge of medicinal plants could be a great combination for this noble quest.
They might have adventures in training Naruto to control the Nine - Tails' chakra better. Jiraiya could take Naruto to different secluded places where he can practice without distractions. Maybe they encounter strange and powerful beasts along the way, which Naruto has to learn to fight using his new skills.
He might encounter unique and dangerous beasts in Uzushiogakure. Fanfiction often has new and exciting creatures for the protagonist to face, and Uzushiogakure's different environment could be home to such beasts.
One possibility could be that Naruto leaves Konoha and stumbles upon a village where everyone talks in rhymes. He tries to communicate but ends up making a comical mess of things. For example, when he asks for food, he says it all wrong and the villagers think he's performing a new kind of comedy act.
Another reason could be that he witnesses corruption or unfairness within the village leadership. If he sees that the system is rigged against certain groups, including himself or his friends, he could grow to hate Konoha and decide to leave.
They could have adventures like learning new jutsus. Maybe they meet some of the famous ninjas in Konoha and get trained by them. They might also explore the unique places in Konoha, such as the Hokage Monument.
Konoha would initially be in shock. Naruto is a hero to many in the village, and his sudden departure with Itachi, who was once considered a traitor, would cause a lot of unrest. The villagers might start to question the leadership of the village, wondering why Naruto left and if there was something they could have done to prevent it.
This type of fanfiction concept is really out of the ordinary. Naruto's character is built on his values of friendship, loyalty, and his dream of becoming Hokage. Killing Sasuke would mean betraying all of that. However, in a fanfiction, the author might be exploring the 'what - if' scenario. For example, what if Sasuke's actions against Konoha were so heinous that Naruto felt he had no choice? But still, it's a far - fetched and very different view from the original Naruto we know.
One possible reason could be to seek stronger opponents outside of Konoha. Naruto might feel that he has outgrown the training opportunities in the village and wants to face greater challenges.
The relationship between Konoha and the Uzumaki clan might be further strained. With Kushina leaving, Konoha would lose an important link to the Uzumaki. And other Uzumaki members might see this as a sign that Konoha is not a safe place for them.