You have a story, a good one. It is in a world that can expand endlessly and support many more stories. That is true of almost every story. Yes, Hamlet and everyone important is dead, but what if we continue telling the story of the aftermath? Some stories are good because they tell of an extraordinary event in an ordinary world. When you keep throwing extraordinary events in, it becomes diluted and unrealistic. Game of Thrones can go on forever because it starts with dragons and zombies. Real world stories don’t have that… hopefully.
(Insert link to zombie apocalypse prepper sponsor here. 🙂
There is nothing sadder than a bad sequel to a good story. Bridges of Madison County and The Notebook had sequels. Count yourself lucky if you didn’t know that. Why do they do it? Money. Or maybe they had a contractual obligation and couldn’t think of anything else. Who knows… The Gone With The Wind sequel at least had the decency to be written after the original author’s death.
The best reason to do it is to follow a side character that everyone wonders about. I wrote 200,000 words of fan fiction about Ayla’s three year old son. The important part of that sentence is “fan fiction”. That is where a story like that belongs. Do you want to write fan fiction about your own story? Mrs. Auel wrote five more fan fiction books after Clan Of The Cave Bear. They had the benefit of being completely different in tone, switching to romance fiction, something that could not be done in the original Ugly Duckling story re-imagining. No one wants to watch a swan getting it on with a duck.
If you want an ongoing series of books, plan for it. Create a larger world in which the later stories make sense. Sherlock Holmes and his fictional detective brethren are perhaps the best form of serialization. Another genre that seems to do this well is the Tech War popularized by Tom Clancy. You start with a reluctant warrior and the world keeps throwing bad guys at them while the technology to wage war becomes ever more lethal. This is just a novelization of the comic book superheroes and their endless adventures against impossible odds.
Serialization doesn’t work well in the Romance world because if they live, do you want to break them up just so they can be HEA with someone else? It is almost antithetical to the genre, which is why the previously mentioned sequels were so bad.
Serialize or not? Make sure your genre supports it. Make sure your characters are interesting and complex enough to support it. Make sure to have character arcs planned out over the series so that you don’t have to RetCon new things for the next story to be interesting. Most importantly, make sure you love these characters enough to spend decades with them. Or… have so many characters that killing them off is the Game that you play. Done properly, it is a great way to just keep writing.