In such a fanfiction, Naruto leaving Konoha could lead to his daughter having a different upbringing. She might be raised in a more secluded place, away from the hustle and bustle of the village. As a result, she could have a more independent personality. And she would probably have to face the challenges of not having the full support of the village that Naruto once had. But with Naruto as her father, she'd surely be strong - willed and determined to make her mark in the ninja world.
He might go on adventures to protect his child from various threats. Maybe he travels to different hidden villages, facing new enemies and making new allies along the way. He could also be on a journey to discover more about his own family's past to better protect his child's future.
Perhaps he would journey to the Hidden Cloud Village. There he could meet new people and learn different techniques. He might also try to bridge the gap between Konoha and the Cloud Village, starting from outside Konoha's boundaries. This would be a way for him to still work towards peace on a larger scale, but away from the restrictions he felt in Konoha.
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.
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.
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.
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.
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.
Well, Naruto might leave Konoha because of the political unrest within the village. He may feel that his ideals are not being supported by the current leadership. For example, if they are more focused on power and control rather than the well - being of everyone. And perhaps he's just tired of the never - ending cycle of conflict that always seems to involve Konoha, and he wants to find his own path away from it all.
Perhaps he feels pressured in Konoha. Having a kid adds more responsibility, and he might think that leaving Konoha will give him the space he needs to grow as a father and a ninja without the constant eyes of the village on him. He could be seeking a place where he can train his kid in a different way, away from the traditional Konoha methods.