One possible reason is that she is in danger. There could be someone or something that wants to harm her or her unborn child, so she has to disappear to stay safe.
Maybe she discovers some dark secret about the father of her child or her family. This knowledge is so shocking that she decides to disappear to figure things out on her own without involving others at first.
It could be a very interesting plot. Maybe Beckett disappears because she wants to protect her unborn child from some unknown danger. Or perhaps there are some secrets related to her pregnancy that force her to go into hiding.
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 they stumble upon a government conspiracy. The government wants to keep something hidden and they are seen as a threat, so they are made to disappear. It could also be related to a scientific experiment gone wrong. They are in the wrong place at the wrong time and get caught in an energy field that transports them to an unknown location, causing them to disappear from their normal world.
Perhaps they are trying to protect their relationship from outside interference. For example, if there are people in Castle's publishing world or Beckett's police department who don't approve of their relationship, they could pretend to be separated to avoid any negative consequences. This would allow them to still be together in private while keeping up the appearance of being single in public. Also, it could be a way to test their relationship. They might want to see how they would fare on their own and if their feelings for each other are strong enough to withstand a period of 'separation'.
Another place to check is FanFiction.net. There are many users who post their own stories there, including those related to 'Beckett Disappears'. Just use the search function to look for it.
He could be influenced by a powerful and mysterious force. This force might be something magical or a very strong jutsu that causes him to lose his memory and wander away from his family. In some fanfics, new and made - up powers can often play a role in such situations. Without his memory, he doesn't recognize his family and just drifts away.
Maybe Alexis has a different vision of the original story or characters. Fanfic often takes creative liberties that might not align with Alexis's view of Beckett.
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.
One great 'Beckett Disappears' fanfic could be 'The Search for Beckett' which focuses on the characters' journey to find Beckett after his disappearance. It has really engaging dialogue.
One possible reason could be for the sake of creating a dramatic plot. The disappearance of Naruto might be used to show how the female characters respond to his absence, highlighting their growth and independence. For example, Sakura might take on more leadership roles in the village without Naruto around.