Well, perhaps Percy could find himself in a dimension where the laws of physics are completely different. He might have to learn new ways to use his powers. For example, if water behaves strangely there, like it's solid in some areas and gas in others, he'd have to adapt quickly.
Maybe it's a corrupted version of one of the gods from Percy's world who has gained even more power in this new dimension. They could be trying to spread chaos and destruction, and Percy has to stop them. This antagonist would know Percy's weaknesses from their time in the original world, which would make the conflict more intense.
Maybe Amy discovers a magical portal in her attic that leads to another dimension. In that dimension, the laws of physics are different, and she has to adapt quickly to survive. There could be strange creatures she befriends or battles against.
In this fanfiction, Percy might encounter different versions of his friends. For example, Annabeth could be a powerful sorceress instead of a strategist. He could also face new and strange monsters that are unique to that universe.
He might encounter new and strange creatures. In many fanfictions, Percy could meet unique magical beings that don't exist in his own world. For example, he could come across sentient plants that can communicate or floating islands with their own mini - ecosystems.
There could be a wise old wizard - like figure in the other dimension who guides Amy. He has lived for centuries and knows all the secrets of this strange place.
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.
In this kind of fanfiction, Harry might encounter new magical creatures that don't exist in his own dimension. For example, he could meet some kind of shape - shifting beasts that can change into any form they like.
When Naruto goes to a parallel dimension in fanfiction, it's a whole new world of possibilities. He could encounter a dimension where his parents are still alive. This would be a huge emotional moment for him. He might also find that the Akatsuki in this dimension has different motives or is even a force for good. Naruto's own values and beliefs would be put to the test as he tries to understand and navigate this strange new place. His usual techniques like the Rasengan might not work as effectively, forcing him to learn and innovate on the fly.