One possible plotline could be that Castle, in his over - the - top way, proposes something like a wild undercover mission where they have to pretend to be a couple in a very improper situation to solve a case. Beckett is initially shocked but then realizes it might be the only way to catch the criminal.
Maybe Castle, in a moment of desperation to keep Beckett close, makes an 'indecent proposal' that involves them going on a long - term trip together under false pretenses just so he can have more time with her. Beckett has to decide whether to go along with it for the sake of their friendship or put a stop to it because it's too unethical. This would cause a rift in their relationship that they would then have to work through.
It could be that an outside force, perhaps a villain, offers Beckett an 'indecent proposal' like dropping a case in exchange for Castle's safety. Beckett is torn between her duty as a detective and her feelings for Castle. And Castle, when he finds out, tries to convince her not to accept the proposal, leading to a complex web of emotions and decisions for both characters.
I haven't actually read this specific fanfic. But generally, a 'Castle and Beckett' fanfic might explore different aspects of their relationship. If it's about an 'indecent proposal', it could potentially be about a situation where one character makes a rather unseemly or out - of - the - ordinary offer to the other, which would create a lot of tension and drama in the story.
One possible plotline could be that the baby has some sort of special ability or is in some way connected to a mystery they have to solve. Maybe the baby's birth coincides with a strange series of events in the city, and Castle and Beckett have to use their skills to protect the baby and figure out what's going on.
It could also be a fanfic that takes place in an alternate universe. In this universe, the events of 4x23 didn't happen as they did in the show. Maybe Beckett didn't become a detective in the same way, and Castle meets her in a completely different situation. They then start to solve mysteries together in this new setting.
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.
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.
Perhaps the man is a new partner Beckett has been assigned to work with on a case. Castle sees them together and his insecurities get the best of him. He starts imagining all sorts of things, like Beckett replacing him in her life. However, in the end, it turns out that Beckett has no romantic feelings for the other man and Castle realizes he overreacted.
In 'Twilight' fanfic, an 'Indecent Proposal' could refer to a plotline where a character makes an inappropriate or unethical offer to another character, perhaps related to power, love, or immortality, in a way that goes against the normal moral code within the 'Twilight' universe.
I'm not sure specifically as there could be many different takes on it. But generally, it might be a fan - made story related to Twilight that involves some sort of inappropriate or unseemly proposal within the context of the Twilight universe. It could be about a character making an improper offer to another, perhaps related to power, love in a twisted way, or some other aspect that goes against the normal moral or social norms in the Twilight 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'.