Perhaps Naruto in the Anbu has a personal mission that goes against the village's orders. For example, he might befriend someone from an enemy village who is actually innocent but is being targeted by Konoha. When the village refuses to listen to his pleas, he decides to leave. He'll use his Anbu abilities to protect his new friend and find a new place where they can both be safe, away from the political drama of the ninja villages.
It could be a rogue Anbu member. This person might be jealous of Naruto's power and popularity within the Anbu. So, they spread false rumors about Naruto which lead to his decision to leave the village. This antagonist would then try to further harm Naruto while he's outside the village, using their own Anbu skills against him.
Well, Naruto could leave the village in a fanfic due to a prophecy. He might believe that leaving is the only way to fulfill a greater destiny that the village can't support. As he leaves, he starts to explore different cultures of the ninja world. He learns about new forms of jutsu and how other villages operate. He could also form a team with outcasts from different villages, and together they strive to change the way the ninja world views things.
Perhaps the ANBU has a personal vendetta against someone in Konoha who is too powerful to take on while remaining in the village. Leaving Konoha gives them the freedom to train and gather allies to come back and face their enemy. They might leave secretly, under the guise of a mission, and start building their strength in the outside world, all while Konoha is left in the dark about their true intentions.
Perhaps Sakura leaves the village because she feels out of place. She might have had some falling - out with her friends or just doesn't feel like she belongs there anymore. So, she packs her bags and sets off on a solo adventure to find a place where she can truly be herself.
In this fanfiction, Naruto might be neglected not just by the villagers but also by some of his so - called friends. This heartbreak makes him leave. He then travels to different lands. Along the way, he meets an old hermit who teaches him ancient ninja techniques that are long forgotten in his village. With these new skills, he contemplates whether to return to the village and show them what they missed out on or to start a new life away from the place that has caused him so much pain.
Naruto might encounter a group of rogue ninjas. At first, they seem like enemies, but as he gets to know them, he realizes they have their own reasons for being outcasts. They could teach Naruto some unorthodox fighting styles that are not taught in the Leaf Village. Also, there could be a mysterious traveler who has knowledge about Naruto's family that even he doesn't know. This traveler could be the key to unlocking some hidden power within Naruto.
The plot could involve an Anbu member protecting the girl who is a Naruto fan. Maybe she has some special knowledge about Naruto that is important in the Naruto world they live in, and the Anbu is tasked with keeping her safe from those who might want to use her knowledge for bad purposes.
In one possibility, Naruto might leave the Leaf due to a misunderstanding. For example, if the villagers continue to show distrust towards him despite his efforts. He could set out on a journey to find a place where he truly belongs. Maybe he meets some new allies on his way, people who understand him without any prejudice.
In some fanfictions, Naruto might leave the Anbu due to internal conflicts within the organization. There could be corruption or power struggles. Naruto, being the idealistic character he is, would not want to be part of such a messed - up system. He might have discovered some dark secrets about the Anbu that made him decide it was not the place for him.
One possible plot could be that Naruto is discovered to have an extraordinary natural talent that is recognized by a powerful figure, like the Third Hokage. This figure decides to fast - track Naruto's training and place him in the ANBU. In the ANBU, Naruto could face discrimination from the other members due to his age and being the Jinchuriki. But he overcomes this with his determination and starts to show his unique abilities, like his unrivaled stamina. He might be sent on missions where his innocent perspective actually helps in solving problems in unexpected ways.