His journey could lead him to the outskirts of the ninja world, to places that are unknown and full of mystery. With the Rinnegan, he might be able to access hidden paths or dimensions. He could be exploring these uncharted areas to gain more power and understanding, away from the politics and rules of Konoha.
One possible reason could be a search for power. The character might feel that Konoha doesn't offer enough resources or training opportunities for them to fully explore the potential of the Rinnegan. They may believe that there are other places in the Naruto world where they can gain more knowledge and power related to their special eye.
Another scenario is that Naruto leaves because he feels like an outcast with the Rinnegan. The villagers might fear his new power, and he decides to leave to find a place where he can fully understand and control the Rinnegan without causing panic. He could then encounter other powerful ninjas or groups who either want to use him for his power or help him master it.
Madara Uchiha could also be the main character in this type of fanfiction. Although in the original story his actions are mostly related to his grand plans for the Ninja world, in a fanfiction, it could be explored what would happen if he decided to leave Konoha specifically because of his Rinnegan and the power it holds.
Well, he could encounter new allies and enemies in different villages and face all kinds of challenges on his own journey.
He might encounter a hermit who lives in a tree and has some really strange habits. The hermit could offer Naruto some 'wise' advice that is actually just really absurd, like telling him to walk on his hands to find his true path.
It could be Sasuke. If he were to turn to the dark side in the fanfiction, his Rinnegan powers could be used to destroy Konoha. He has the potential to be extremely powerful and if he has some sort of vendetta or is influenced by some dark force, he might take his anger out on the village.
On a more positive note, it could inspire some young ninja to step up. They would see Naruto's departure as a challenge to prove themselves and fill the gap he left. His leaving could also make Konoha re - evaluate its relationship with its ninja and how to keep them satisfied and in the village.
It could lead to political unrest in Konoha. There might be those who blame the village leadership for Naruto's departure. This could lead to power struggles as different factions try to take control and change the way the village is run. Some might demand more transparency and better treatment of their shinobi to prevent such a thing from happening again.
Konoha's power structure could be affected. Naruto is one of the strongest shinobi in the village. Without him, the village might be more vulnerable to attacks from other villages or threats. They would need to quickly find ways to fill the gap in their defenses.
It could create a sense of unease and fear. Naruto is a very powerful ninja, and if he turns dark and leaves, the villagers would be worried about him coming back as an enemy. It might also lead to a decrease in morale among the ninjas who once considered Naruto a friend or ally.