Aizen might be motivated by power. Ichigo has a very unique power set that Aizen may want to harness for himself. And Rukia, being from Soul Society, could be used as a way to gain information or leverage against Soul Society.
One idea could be that Aizen kidnaps them to use Ichigo's power and Rukia's knowledge of Soul Society for his own dark plans. He might want to study Ichigo's unique Hollow - Shinigami powers up close and use Rukia as a bargaining chip with Soul Society.
You can try searching on popular fanfic platforms like Fanfiction.net or Archive of Our Own. Just use the keywords 'Aizen - Obsessed Ichigo Rukia' in their search bars, and you might find relevant fanfics.
One of the things that can make it interesting is the exploration of different character arcs. Aizen's obsession can be a catalyst for Ichigo and Rukia to grow in ways they might not have in the original story. It can also bring out different sides of their personalities. For example, Ichigo might become more determined to protect Rukia from Aizen, showing a deeper level of his loyalty. Rukia, on the other hand, could develop new strategies to deal with Aizen's threat, highlighting her intelligence and resourcefulness. Additionally, the fanfic could introduce new settings or concepts related to Aizen's obsession, such as alternate dimensions or new forms of power, which can keep readers engaged.
Maybe Aizen has created a false asylum as part of a new experiment. He lures Ichigo and Rukia into it, and they have to find a way to escape while also dealing with the illusions and traps Aizen has set up. In the process, they could learn more about Aizen's motives and past, and perhaps even find a way to defeat him once and for all.
One possible storyline could be a story where Ichigo and Rukia have to team up against Aizen's new scheme. They gather a gang of their friends and allies from the Soul Society and the human world. They train together, combining their powers in unique ways to face Aizen's powerful minions. Along the way, there could be some character development for each member of the gang, like Ichigo learning a new form of his Zanpakuto through the guidance of an old Soul Reaper in the group.
The combination of these characters. Aizen is a complex and powerful villain, Ichigo is the determined hero, and Rukia has her own unique charm. When you add Aslem into the mix, there are so many possibilities for different storylines. It could be about power struggles, or about forging new alliances, which is very appealing.
You can start by looking on fanfiction platforms like FanFiction.net or Archive of Our Own (AO3). These sites have a vast collection of fanfics related to 'Bleach' which might include the characters Aizen, Ichigo, and Rukia in asylum - related scenarios. Just use the search function and type in relevant keywords.
In a fanfic, Ichigo and Rukia's relationship could be developed by introducing emotional arcs. Maybe Rukia has a secret she's been keeping from Ichigo, and when it's revealed, it causes a rift between them. But through their shared love for their friends and the Soul Society, they eventually overcome it. Aizen could be the catalyst for this. His actions could force Rukia to reveal her secret. Meanwhile, Aizen's relationship with them could be developed by showing his envy or admiration for their relationship. He could try to break them apart because he sees their bond as a threat to his own plans, or he could be secretly longing for a relationship like theirs, which adds a new layer of complexity to his character.
Rukia is also very popular. Her cool and composed nature, along with her powerful ice - based Zanpakuto abilities, make her stand out. Her backstory, being from the Soul Society and her connection to Ichigo, gives writers a lot to work with in terms of character development and plot.
I'm not sure which specific one to recommend off - hand, but you can try looking on fanfiction.net. There are likely many 'Aizen Aslem Ichigo Rukia fanfic' there. You can search by tags like 'character relationships' or 'action - adventure' to find ones that might interest you.