She could find herself in a world with different rules. Like gravity might work in a strange way or the environment could be made of substances she has never seen before. It's also possible that she meets new characters who are either friendly or hostile, and she has to figure out how to interact with them.
Well, she might encounter different versions of her friends. Maybe a Sonic who has a completely different personality or abilities. For example, he could be a dark Sonic in that alternate dimension.
She could meet an alternate version of herself. This other Amy Rose might have different values or goals.
She might encounter strange creatures. For example, in many alternate dimension stories, there are unique beings that don't exist in our world. Maybe she'll meet a talking animal or a floating jelly - like entity.
She might run into a group of rebels. If the alternate dimension has some sort of oppressive regime, there are likely to be rebels fighting against it. Amy could get involved with their cause and help them in their struggle.
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.
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.
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.
Even one of her friends like Tails could be the betrayer in a really unexpected plot twist. Maybe Tails is forced to betray her by some external threat. For example, if an enemy captures Tails and threatens to harm him or Sonic if he doesn't give away some important information about Amy Rose or lead her into a trap.
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.
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.
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.