Seeds everywhere. Gets purple flowers. It's a weed to me.
I'd definitely say it's a columbine. Usually a short lived perennial flower and, as you've seen, it self-seeds readily if you don't dead-head it. If you don't like it where it's growing, it's a weed. If you might like it better in another place you could transplant it. In my experience they handle transplanting fairly well.
If extermination is your goal, plan on weeding in spring, that's when the new seedlings come up, if you get them early they never have a chance to get root reserves to carry them through your attacks.