Hor pepper plants can be overwintered. Last year, I overwintered a Jalmundo (jumbo Jalapeño) plant. To do that, dig up the plant. This one is a Cayenne pepper that has been harvested.
Now hose off all of the soil. That removes insects and other harmful creatures.
Now prune the roots and top to a more compact shape. The top pruning can be pretty radical. I pruned to about 1/2" above nice looking nodes. Each node can produce several branches when it starts growing again.
Now plant the pruned hot pepper plant in a clean container with good quality potting soil. Water it in. Let it drain. Now store in bright, cool, frost-free room.
Here it is with two friends, a Jalapeño and a Banana pepper plant. Also a dendrobium orchid.
These will be allowed to dry out, with only slight moistening until late winter. Now they are in the garage. I will move them to a cool room, before it's too cold.