Well, one possibility is that Harry would be exposed to a whole new world of magic. Dr. Strange's magic is very different from the magic in the Harry Potter universe. Harry might learn new spells and ways of using magic that could make him even more powerful. He could also get involved in Dr. Strange's battles against otherworldly threats, which would be a huge change from his battles with Voldemort.
Harry's social circle would expand in a very different way. He'd be introduced to the other sorcerers and strange beings that Dr. Strange knows. This could be both exciting and a bit intimidating for Harry. He might also have to deal with new enemies that come from the threats in Dr. Strange's world, which would be a whole new level of danger compared to what he faced at Hogwarts.
The magic could blend in interesting ways. Harry's wand - based magic might be enhanced by Dr Strange's more esoteric and hand - gesture - based magic. For example, Harry could use his wand to direct the energy patterns he learns from Dr Strange.
It might not be a traditional or 'legitimate' in the sense of being part of the original canons. But in the realm of fanfiction, it's a great idea. Fanfiction is all about creativity and mashing up different universes. Harry Potter has its own set of magic rules and Doctor Strange has his. Blending them could result in a whole new set of magic spells and powers that could be really cool to explore.
Hermione Granger and Dr Strange might also be a popular pairing. Hermione's thirst for knowledge and intelligence would pair well with Dr Strange's vast knowledge of the mystic arts. They could collaborate on researching new spells or ways to protect their respective worlds.
A possible crossover could involve the characters' moral compasses. Harry Potter has a strong sense of right and wrong based on his experiences at Hogwarts. Dr Strange, on the other hand, has a more complex view of morality due to his dealing with different dimensions and powers. A fanfiction might explore how they interact when faced with a difficult moral decision, like whether to sacrifice one life to save many.
If Harry Potter was raised by Dr. Strange in fanfiction, it could lead to some really interesting character developments. Harry might be more confident in handling strange and unknown magical situations, as Dr. Strange is constantly dealing with the unexpected. Also, he may have a different set of values. Instead of the house - based values at Hogwarts, he could be more focused on the balance of the multiverse like Dr. Strange.
Well, it could be that Dr Strange uses his magic knowledge to train Harry Potter in different ways of spellcasting. Maybe he shows Harry how to manipulate time and space while casting spells, which could give Harry an edge in fighting Voldemort. He might also teach Harry about different magical dimensions and how to draw power from them.
In this kind of fanfiction, one possible scenario is that Harry's whole view of the wizarding world would be warped. Voldemort might try to mold Harry into his own image. He could teach Harry dark magic instead of the traditional magic taught at Hogwarts. Harry might grow up with a completely different set of values and powers, which would surely change the whole dynamic of the fight against good and evil in the wizarding world.
In this kind of fanfiction, Harry might grow up in a very different environment. Poppy is a healer, so she could potentially raise Harry with a great deal of medical knowledge. He might learn about healing spells and potions from a very young age. Also, Poppy's kind nature would likely give Harry a more nurturing upbringing compared to his life with the Dursleys.