One common theme is self - discovery. Naruto often has to re - evaluate who he is and his abilities in the new time. Another theme is changing the future. He may try to prevent bad things from happening in the alternate time that he knows happened in his own. Also, the theme of adapting to new cultures and ways of life is prevalent as the alternate time usually has different ninja traditions and social structures.
He could meet an alternate version of himself. This Naruto might have made different choices in life, like choosing not to pursue being a Hokage. It would be interesting to see how they interact.
One popular 'Naruto goes to alternate time' fanfiction is 'Naruto: Time Traveler'. In this fic, Naruto is sent to an alternate time period where he has to face new challenges and enemies. Another one could be 'Naruto in a Different Era' which explores how Naruto's presence changes the course of history in the alternate time. There's also 'Naruto's Alternate Time Adventure' that focuses on his journey of self - discovery in the new time.
In the alternate dimension, Naruto could find that the Akatsuki are actually the protectors of the villages. This would completely change his view of the world. He would have to re - evaluate his ideas about good and evil and decide whether to help them or oppose them depending on their true motives.
Perhaps in this alternate reality, there are no ninja villages at all. People live in a more unified world, but they face a new and powerful enemy that Naruto has to help fight. He would have to adapt his skills and learn new ways of combat that are unique to this world.
He might meet alternate versions of his friends. For example, Sasuke could be a completely different person with different goals and a different personality. Maybe he's a peaceful scholar instead of a revenge - seeking ninja.
Well, in some 'naruto goes to alternate universe fanfiction' plots, Naruto might find himself in a world where all the ninja clans have merged into one big super - clan. He'd have to navigate the complex politics within this new structure. Also, there could be a plot where he enters an alternate universe where the Fourth Great Ninja War never happened, but there are new threats emerging that he has to face. And then there's the idea of a universe where Naruto's parents are alive, and he has to deal with a completely different family dynamic.
Perhaps he'd meet a different Hokage. One who has led the village in a completely different way than in his own dimension. Naruto could learn a great deal from this Hokage about leadership and different ways to protect the village.
Sasuke is often a common character. Since they have such a deep relationship in the original Naruto story, it's likely they'll interact in the alternate universe too. Kakashi could also be a common one. He was Naruto's teacher, so it makes sense for him to be around in different universes. And Sakura, as part of the original team, is also likely to interact with Naruto in these fanfictions.
His chakra might function differently. In the alternate reality, the rules governing chakra could be altered. For instance, his Nine - Tails chakra could have a different color or effect.