He might leave Japan in the fanfiction because of some threat or danger that he believes can't be resolved there. So, he ventures out to find a solution away from his homeland. This could lead to him making new allies and discovering new powers or techniques in a different environment.
In this fanfiction, Izuku could leave Japan to follow a lead on his father or some family mystery. He's always been curious about his family, and leaving Japan could be the start of a grand adventure to uncover the truth. He might meet people who knew his father along the way and learn things that change his perspective on being a hero.
One could be All Might. Even if Izuku leaves Japan, All Might might still be in contact with him, guiding him from afar. Another could be Bakugo. Their relationship is complex, and Bakugo might have some sort of reaction to Izuku leaving, perhaps feeling a sense of rivalry still and trying to prove he can be a great hero in Japan while Izuku is away.
One possible plot could be that Izuku suddenly finds himself with Loki's powers and memories. He has to deal with the newfound ability to deceive and manipulate, which goes against his usual honest nature. He might use these powers to outwit villains in a different way than he usually does in his own story.
Well, it could be about Izuku being an alien. Maybe he has some super - strange powers that are not like the normal quirks in his world. For example, he could have the ability to manipulate time or space, which would make him really different from other characters.
Maybe it's about Izuku using his wealth to support his hero training in unique ways. He could afford the most advanced support items and training facilities.
Well, it could be about Harry going to Japan for a magical exchange program. Maybe he meets new wizards there, discovers unique Japanese magical creatures, and learns different types of magic that are specific to Japan.
In some fanfictions, Aizawa might be under the influence of a villain's quirk that makes him act against his will. So, he kidnaps Izuku, but later on, he realizes what he's done and tries to make amends. This could lead to a story full of drama and redemption.
Well, one possible plot could be that his friends have to take care of him. They might struggle at first, not knowing how to handle a baby Izuku. For example, Bakugo could be really frustrated but gradually learn to be gentle. Ochaco might be really good at soothing baby Izuku when he cries.
Well, it could be that Monoma is constantly taunting Izuku about his quirk or his status at school. Izuku, usually patient, finally snaps under the pressure of Monoma's relentless jabs. Maybe Monoma makes a comment about how Izuku got into U.A. through some sort of fluke, and Izuku just can't take it anymore and lashes out verbally or even physically.
The plot could involve a tournament - style scenario where heroes and shinigami from different universes are gathered. Ichigo and Izuku are pitted against each other in the first round. However, instead of fighting, they realize that there is a greater conspiracy at play. Some powerful entity is trying to use the tournament to drain the powers of all the participants. So, they decide to work together, combining Ichigo's zanpakuto skills and Izuku's One For All quirk to expose and defeat this hidden enemy.
In this fanfiction, perhaps Izuku's pet is a magical being. It could be that the pet was sent to Izuku by some mysterious force to help him on his journey to become a great hero. The pet might be able to sense danger and warn Izuku in time. As Izuku grows closer to the pet, he discovers more about its past and the secrets it holds. This relationship could also attract the attention of villains who want to use the pet's power for their own evil purposes, leading to some intense battles and rescues.