She might become more independent. Without the structure of Hogwarts, she'd have to figure things out on her own. For example, she'd have to find her own sources of magical knowledge. This could make her a bit of a maverick in the wizarding world.
Her relationships would be different. Instead of having school friends like Harry and Ron, she might form alliances with other self - taught wizards or those from different magical backgrounds. This could lead to a more diverse and complex social network, which in turn would shape her character in unique ways. For instance, she could become more diplomatic as she deals with different magical cultures outside of the Hogwarts - centric wizarding world.
She could end up in a non - magical educational institution for a while. During this time, she might try to apply her love for learning in a different context, like studying advanced sciences. However, her magical abilities would still be there, and she could end up using them in unique ways, perhaps accidentally, which could draw the attention of the wizarding world and lead to her being brought into the magical community in a very different way than going to Hogwarts.
The plot might develop with Harry exploring the non - Hogwarts magical world. He could meet new characters who introduce him to different magical cultures and spells.
Her character could develop through her relationships. She might befriend students from all houses, which is a bit unusual in Hogwarts. This would be because she has qualities from both Gryffindor (from Hermione) and Slytherin (from Snape). Through these friendships, she'd learn different perspectives and grow as a person. For instance, she could teach her Gryffindor friends about the subtleties of Slytherin thinking and vice versa.
Her character might develop in a way that she constantly tries to prove herself. Because of her parents' polar - opposite reputations, she would feel the need to show that she is not defined by their actions or reputations. She could excel in both academics and magic - related sports, like Quidditch, to gain respect.
Hermione's character might develop a greater sense of adventure. Usually, she's more into academic things in the original story. But in the fellowship, she would be exposed to different cultures and terrains. This would make her more open - minded and adventurous. She could also learn to be more flexible in her thinking, as the fellowship's situations are often unpredictable.
Hermione's character would likely become more laid - back. Growing up in the Weasley household, she'd be exposed to their carefree and fun - loving nature. She might not be as uptight about rules and studies all the time. For example, she could start joining in on the twins' pranks more often.
Hermione's character would likely be more complex. The founders had their own quirks and beliefs. If she was under their influence from a young age, she would inherit some of those traits. For instance, if raised by Salazar Slytherin, she might have a more calculating side, while still maintaining her good nature. This combination would make her a very interesting and different character compared to the Hermione we know in the original books.
If Hermione joins the BAU in fanfiction, her character would develop in many ways. Firstly, her knowledge base would expand as she learns about forensic science and modern - day criminal psychology. Her magical knowledge would start to blend in unique ways with her newfound skills. Emotionally, she'd become more resilient. Dealing with the horrors of some of the cases would toughen her up. Socially, she'd learn to communicate in a more direct and less academic way, which would be a big change for her. She might also start to see the world in a more grey - area way rather than the more black - and - white view she had in the wizarding world.
She would become more confident in a non - magical combat situation. The Avengers' training and battles would force her to learn new skills and be more assertive physically. For instance, she might start learning martial arts from Captain America.