It could be that Castle is overprotective. He loves Alexis so much that even the slightest sign of Beckett not being on the same page as Alexis in Castle's eyes makes him think Beckett hurt her. Maybe Beckett gave Alexis some career advice that Castle thought was too pushy.
She could be defensive of Beckett. Since Alexis probably has a good relationship with Beckett, she may try to convince Castle that he's wrong and that Beckett didn't mean any harm. For example, she might tell Castle about all the positive things Beckett has done for her.
Maybe he saw Beckett spending a lot of time with another man without knowing it was for a case. Castle is a bit of a jealous type, and he might overreact to normal interactions.
Maybe in the fanfic, Beckett got a new job opportunity in a far - away place. This could make Castle think she's leaving him.
Love and acceptance is also a prevalent theme. Since Beckett comes into Castle's life and becomes part of Alexis's world, fanfics may focus on how they all learn to love and accept each other fully. This could involve storylines where there are initial misunderstandings but eventually they come to a place of complete acceptance and harmony in their relationships.
In the fanfic, it could be that there are signs that Beckett is hiding something. Perhaps he has unexplained absences. He might say he's at work but Alexis can't reach him during those times. Also, he could be overly protective of his phone or personal space, which makes Alexis suspect that he's cheating.
There are definitely some great 'castle beckett and alexis fanfic' pieces. If you like stories that explore the emotional side, you might find some where Alexis has to deal with the dangerous aspects of her father's and Beckett's job. Others could be more light - hearted, showing their day - to - day interactions in a humorous way.
He might be in denial at first. He could keep trying to convince himself that there's no way Beckett would leave him.
One possible reason could be that they are on a secret mission. Maybe they are trying to protect a key witness or important evidence from a powerful and dangerous criminal organization. In the world of fanfic, this could lead to them having to disappear to keep their whereabouts unknown.
Maybe he's afraid of losing her because she's his partner and they've been through so much together. Their relationship is very important to him.
Fanfiction.net is a great place to start. It has a large collection of fan - created stories, including ones related to Castle, Beckett, and Alexis.