One idea could be that in the alternate dimension, Voldemort won the war long ago. Hogwarts is now a dark and dreary place, with Death Eaters as teachers. Harry and his friends, who are part of an underground resistance, have to find a way to travel back in time to change the course of history.
Perhaps in this alternate dimension, magic is not hidden from the Muggle world. Instead, there is a co - existence. However, this has led to many problems. The Ministry of Magic has become a sort of global superpower, trying to control both magical and Muggle affairs. Harry and his friends could be involved in a plot to restore the balance between the two worlds and prevent the Ministry from becoming too tyrannical.
One idea could be that in the alternate dimension, Voldemort won the war and Hogwarts is now a dark and dreary place where students are taught only dark magic. Another could be that the wizarding world is merged with the Muggle world in a strange way, like there are magical creatures running amok in London streets. Or perhaps in this alternate dimension, Harry was sorted into Slytherin and has to deal with a whole different set of challenges and prejudices.
One idea could be that in the alternate dimension, the Batfam members have swapped personalities. For example, Batman is overly optimistic and carefree while Robin is brooding and serious. Another plot could involve an alternate Gotham where the city is a utopia, but the Batfam has to deal with a new kind of threat that thrives in this perfect environment. Maybe a villain who wants to bring chaos to this idyllic Gotham.
One idea could be that in the alternate dimension, Dementors are actually friendly creatures that help protect Hogwarts instead of causing fear. They could use their powers to keep dark wizards at bay in a different way.
One idea could be that in this alternate universe, Hufflepuff is the dominant house in Hogwarts. They have a secret society within the house that holds the key to an ancient magic that can save the wizarding world from a new, unknown threat. And the main character, a Hufflepuff student, discovers this by accident and has to take on the responsibility to gather the members of the society.
One interesting idea could be that Harry decides not to defeat Voldemort in a traditional battle. Instead, he uses a powerful spell of love and redemption to turn Voldemort back to his original self, Tom Riddle, a lost and confused boy. This would completely change the whole dark - vs - light dynamic of the story.
One idea could be that Harry has an alternate identity as a Muggle detective. He uses his magical skills secretly to solve cases that have a magical element but are disguised in the Muggle world. For example, he might be investigating a series of strange disappearances that are actually the work of a dark wizard trying to gather ingredients for a powerful spell.
Well, a cool plot for 'Harry Potter in Another Dimension' fanfic could be that Harry discovers he has a twin in this other dimension. His twin has been raised by Voldemort and has a very different view of the wizarding world. Harry has to find a way to convince his twin to abandon Voldemort's ways. Or, what if in this dimension, the Sorting Hat can talk out loud all the time and gives very different sorting decisions? For example, it might put all the brave students in Slytherin and the cunning ones in Gryffindor. It would be really interesting to explore how the relationships between the houses change.
A really cool plot could be that the traveler goes to a dimension where magic is fading away. They discover that it's because of an ancient curse. So, they embark on a journey to find the origin of the curse and a way to break it, all while interacting with alternate versions of the well - known Harry Potter characters.
Well, perhaps in some 'Harry Potter dimension jumper fanfiction', there's a dimension where the Marauders are still alive and they team up with the modern characters to fight a new dark threat. Or a dimension where magic is dying out and the jumper has to find the source of the problem and fix it. This could involve traveling to different magical locations in that dimension, like the Forbidden Forest which might hold some secrets.
One idea could be that Peter discovers he has superpowers unique to this alternate dimension. For example, he might be able to control time in a different way than in his own world. Another plot could involve Peter meeting alternate versions of his friends and family, and having to deal with the differences in their personalities and relationships. Maybe in this dimension, his best friend is his enemy, and he has to figure out how to change that.