Another theme is redemption. Fanfic authors like to explore the idea of Akutagawa making up for his past wrongs. He might be given a chance to save someone or change an event that he regrets in his past. It gives a new dimension to his character.
One great BSD Akutagawa crossover fanfiction is 'Akutagawa in Another World' which explores what would happen if Akutagawa was transported to a different universe. It has an interesting plot where he has to adapt to new powers and a new society.
One popular theme is the development of their relationship from rivalry to something more complex. They start as enemies in the original story, so fanfictions often explore how they might gradually understand and care for each other.
Archive of Our Own (AO3) is a great place to find such fanfictions. It has a large and diverse collection of works, with various tags and filters to help you narrow down your search to exactly what you're interested in regarding Atsushi and Akutagawa.
In atsushi time travel fanfiction, Ranpo can be a great character to pair with Atsushi. Ranpo's super - detective abilities can play an important role when they are in different time periods. Chuuya is also a popular choice. His powerful ability and his connection to the Port Mafia can add a lot of drama to the story. And Yosano is another one. Her ability to heal and her strong personality can make for some interesting interactions with Atsushi during time travel.