I watched three episodes of Person of INterest because the premise was interesting, and concluded it was just another boring CBS-procedural with an awful, awful leading performance (Caviezel).
I've heard it's gotten better, but it's really hard for me to come back to stuff I've given up on (see : Marvel's Agents of SHIELD).
It's not just gotten better, it's gotten incredible. I can't remember when it happened though. If you were interested, I could find approximately when it stopped being a mediocre case-of-the-week procedural and became an awesome serialized sci-fi dystopian drama.
I agree it has become incredibly well done. My wife was really excited about it before its debut, and I was not. It hooked me immediately, and she let it go quickly. It has so much going for it - high quirk factor, better than average acting even when scenery is being chewed, lighting effects, micro vs. macro morality. I like it a lot, and am behind a few episodes.
Person of Interest is a very good show, and all the episodes are fantastic. That being said, I know a few people who don't like the way the show is headed, they prefer it just the way it was, with them saving or stopping the new number every week. I like both aspects, but I'm slightly inclined to agree with the unmentioned individuals.
Even though I enjoy POI, I really like Arrow and Flash now. One Saturday night, my friend and I randomly watched the Arrow pilot after we had heard that the creators of something called "Arrow" were doing "Flash". Flash sounded pretty good, but we were wondering what Arrow was. So we watched the pilot on a Saturday. 2 hours later, 2 more friends come over, asking what we were doing. We excitedly responded, " We just watched this thing called Arrow. We're in the middle of the 3rd episode, would you want to join?" We ended at about 11:30 P.M. Right now, I believe Arrow is the best show out there. Haven't seen Gotham yet, but it sounds good.