Kaguya otsutsuki is often a very popular character. Since she was the progenitor of the otsutsuki in the Naruto canon, fanfictions like to explore more about her backstory and her relationship with other characters. Another popular one is Momoshiki otsutsuki. His powerful abilities and his encounter with Naruto in the series make him a great character to expand on in fanfiction.
Kaguya Otsutsuki is often a central character. She has a significant role in the original Naruto lore related to the Otsutsuki, and in fanfictions, her backstory might be further elaborated or her powers could be enhanced or explored in different ways.
Momoshiki Otsutsuki is also quite popular. His powerful abilities and his role as a major antagonist in the Boruto series give a lot of material for fanfic writers. They can expand on his backstory, or create alternate scenarios where he has different interactions with the main characters.
Kaguya Otsutsuki is often very popular. As the progenitor of chakra, she has a lot of mystery surrounding her, and fanfiction writers like to explore her backstory further, perhaps giving her more complex motives for her actions in the original story.
Indra and Asura are also characters that are frequently paired with Hagoromo in fanfiction. These pairings often explore the family dynamics more closely, such as Hagoromo teaching them new skills or them helping Hagoromo in some way that was not shown in the original series.
Saitama from One Punch Man would be an interesting pair. Saitama's simple yet overpowered nature could create an interesting dynamic with Hagoromo's complex and ancient powers.
A fanfic could focus on Indra's training. We could write about the intense and arduous training he underwent to become such a powerful figure. It might involve secret techniques passed down in the Otsutsuki clan that only he was able to master due to his unique talent.
Another angle could be a fanfic where Indra Otsutsuki travels to a different dimension. He has to adapt to a completely new world with different rules and powers. It would be fascinating to see how his Otsutsuki abilities fare in this new environment and how he interacts with the local inhabitants.
Yes, there are likely some fanfictions with this concept. Fanfiction writers often come up with unique and creative ideas, and the idea of Naruto having some sort of special connection or relationship with Kaguya Otsutsuki in a fan - created story is not unheard of. It could involve Naruto exploring Kaguya's past, or perhaps having a role in her redemption in the fanfic's alternate universe.