Well, it could go in many directions. Maybe Naruto starts a quest to find a way to bring peace to the ninja world on his own terms. Or perhaps he has to deal with the consequences of leaving Konoha and tries to prove his intentions were pure.
Well, it could be that Naruto leaves because he realizes that Konoha's ways don't align with his ideals anymore after that intense mission. Or perhaps there are hidden threats that force him to go away for the greater good.
One reason could be that he feels a sense of disillusionment. After the war, he might see that the village he fought so hard for still has deep - seated problems and prejudices. He might think that leaving is the only way to find true peace for himself. Another reason could be to protect the village from any potential threats that might target him. If he's away, those threats might not come to Konoha.
Maybe there are some dark secrets within Konoha that he discovers in the fanfic. These secrets could be related to the treatment of certain individuals or something more sinister about the village's leadership. Discovering such things could make Naruto decide to leave in order to distance himself from the corruption or unfairness.
Sakura would probably be very worried. She has always cared about Naruto deep down, so she might be constantly trying to find a way to bring him back. Kakashi, on the other hand, would likely understand Naruto's reasons to some extent. He might be sad but also respect Naruto's decision. As for Sasuke, if he is in the village at the time, he could be conflicted. He and Naruto have a complex relationship, and Naruto's leaving might make him realize how much Naruto actually means to him.
Naruto himself might start to form his own group or alliance outside of Konoha. He could gather other outcasts or those who have a grudge against the major ninja villages. This new group could then have its own goals, like trying to reform the ninja world in a way that is different from how Konoha and the other villages operate. As the story progresses, there would be inevitable clashes between Naruto's new group and Konoha.
One possible reason could be for Naruto's safety. Maybe she has to go on a dangerous mission and believes Konoha is the safest place for him. Another reason might be that she is forced by some powerful enemies who threaten Naruto's life if she doesn't leave him in Konoha.
In a 'naruto and kushina leaves konoha fanfic', perhaps they leave due to a threat to Naruto's safety that the village can't fully protect against. Maybe some secret organization is after him and they feel they have no choice but to go into hiding outside Konoha.
In a 'Naruto Namikaze leaves Konoha' fanfic, it could be that Naruto is framed for a crime he didn't commit. The villagers turn against him, and even his friends start to doubt him. Feeling betrayed and alone, he decides to leave Konoha. As he travels, he meets other outcasts who have also been wrongly accused, and together they form a group to clear their names and seek justice.
One idea could be that as Naruto leaves Konoha, he encounters a powerful and ancient creature in the wild. In a battle with it, the extreme stress and need to survive unlocks his hidden bloodline. This new power could be related to some kind of elemental control that is unique to his clan, like a special form of wind manipulation that is far more powerful than what he had before.