She could be a vigilante like her father, but with a more flamboyant style inspired by her mother. Imagine her swinging through Gotham in a purple and black catsuit, using her wits and combat skills to take down criminals.
Her story might involve trying to bring her parents closer together. Since Batman and Catwoman have a complex relationship, their daughter could be the catalyst for them to finally have a more stable and loving relationship. She could see the good in both of them and work towards making them a family.
Well, she could be a character who struggles with the duality of her heritage. On one hand, she has Batman's sense of justice and order, but on the other, Catwoman's more fluid view of right and wrong. This internal conflict could be a major part of the fanfic. She might also have her own love interests, perhaps a character who is neither fully on the side of the law like her father, nor a thief like her mother, adding another layer of complexity to her story.
Maybe the daughter discovers a secret conspiracy in Gotham that involves both her parents' past enemies. She has to use her unique combination of skills, like Batman's detective work and Catwoman's acrobatic abilities, to uncover the truth. Along the way, she forms her own team of allies, some from the side of the law and some from the fringes.
In this fanfiction, their daughter might inherit Batman's sense of justice and Catwoman's agility. She could grow up torn between the two worlds of her parents. Maybe she'd use her skills to fight crime in a different way than Batman, with a bit more of Catwoman's slyness. For example, she could use her charm to get information from villains before taking them down.
The Penguin could be a main antagonist. He might see the daughter as a new threat to his criminal operations in Gotham. Since she has the combined skills of Batman and Catwoman, she could easily disrupt his illegal business, like his smuggling operations.
In some fanfiction, their daughter could be a powerful force for change in Gotham. She might use her parents' influence and resources in unique ways. For example, she could start a new movement to clean up the city, using her mother's charm and her father's detective skills to expose corruption and bring about real change. This would be a fresh take on the Batman - Catwoman story and add a new layer of depth to the Gotham universe.
Maybe Batman gets jealous when Catwoman teams up with another anti - hero. He's so used to being her main partner in crime - fighting (sort of), and this new alliance makes him feel left out. It would be an interesting angle to explore how his jealousy affects their dynamic.
One possible plot could be that Batman and Catwoman have a child who discovers their parents' secret identities by accident. The child might then try to use their own skills, a combination of Batman's detective skills and Catwoman's agility, to help their parents fight a new villain in Gotham.
She could be different in her appearance as well. Instead of Batman's all - black and intimidating look or Catwoman's skintight black suit, she could have a costume that is a blend of colors, symbolizing her unique position between her parents. Emotionally, she could be less brooding than Batman and less flirtatious than Catwoman. She might have a more empathetic side, trying to understand the motives of the criminals she faces rather than just punishing them.
Maybe a plot twist could be that they start to understand each other on a deeper level. Batman realizes how difficult it is for Catwoman to constantly be on the run from the law, and Catwoman sees the burden Batman carries to protect Gotham. This new - found understanding could change their relationship in the long run.
She could become an ally of Batman. Despite her father's villainy, she might see the error of his ways and decide to help Batman fight crime in Gotham. This would be a huge twist in the Batman - Penguin dynamic.