That is clearly a mantis egg case. Keep it in a cool, dark place over winter, then when the sun warms things up in spring, take it outdoors and hundreds of tiny mantises will emerge. These are valuable beneficial insects, and people pay a lot of money to buy them. It is best if you do not remove the egg case from the pepper branch.

I find them nearly every fall on my pepper plants, which seem to be one of their favorite places to leave eggs. One year, I took an overwintered case and placed it in a fishbowl with screen over the top. In May, the little rascals emerged in large numbers, looking for all the world like green mosquitoes. I released them out in the orchard.

Don Yellman, Great Falls, VA