How and when to split a hive

I presently have a very healthy and productive hive.  I am wondering about starting a new hive next spring  by splitting this one. How and when should I do this?

if the hive is heathy and productive a gang of bees inside will produce a new queen and split off on their own, you need to watch and before the scouts find a wild spot to make new hive capture the ball of bees into a box. they usually gather on a branch or something not far from the hive they split off of while the scouts are looking for a place to protect the queen. does this make any sense? there's a ton of good free info on beekeeping on internet you might want to learn as much as you can before jumping in with both feet.
a lot of the colony collapse disorder involves bees that are transported from one place to another to aid pollination of monoculture crops. but a nest of wasps can destroy a struggling honey bee hive in short order.

I have raised bees on and off for decades. In the past, I have purchased bees .... except for the occasional piece of dumb luck where the swarm adopted an empty hive.  I am wondering how to ID the issue before the swarm..... and how to make the transfer.

It seems you're talking about two different things. Splitting a colony is different than swarming.

There are several methods for making splits. A blind split is simply separating two full boxes of bees & adding some bottom & top covers to one. It may or may not work. It mostly depends on if the box without the queen has very fresh eggs (2 or 3 days old maximum) so the worker bees can make a new queen.

A more reliable method is to find the queen & make sure the queenless box that is removed has very fresh eggs.

Probably the most reliable method for splits is to buy a mated queen for the queenless box.

It's getting toward the end of season to be making splits this year. Might not be enough drones available for her to mate with.

The picture shows swarm cells ... new queens being made. If you see those a swarm is imminent. You can wait for them to swarm & hope to capture them. Or remove a frame containing those plus about half the bees to make a new colony. Be careful with that method because they might be supercedure cells which means the old queen is being replaced for whatever reason. She might already be dead or injured or just not laying good enough.

I am not planning on taking any action until spring.

My thinking is that splitting the hive will reduce the probability of swarming.

Yes, my assumption was that buying a queen was the best bet. I wanted to see what the other options might be.