Pretty cool what they did there with all the photographs (though none of them look like they are actual pictures of something, they're merely tailored to look like his profile when combined; maybe the finished product will fix that oversight.) So maybe there is some overarching mystery hanging over her head for this story arc instead of having a few different cases to solve like in Daredevil where first it was the bowling alley murder, then the housing takeover, then the four main leaders of the gang Fisk ran being chipped away at until there was just him left (and his accountant was bumped off too). Is she looking for the Purple Man for revenge or is it something else? Obviously she's haunted by him and what he put her through. Is he after her or is he just counting on her maybe finding her way back to him somehow? I like how because there's no way of knowing how closely it will follow the comics that it leaves so many questions, we don't know automatically what does happen with the story.
It kind of makes sense how Kilgrave could go undetected, but then it kind of doesn't. Anyone who comes after him is under his control so they can be influenced to see something completely different, so just identifying him could lead them on a wild goose chase depending on what he chooses to make them see. He can make them commit a massive slaughter or just kill themselves, putting his hunters in the hot seat. The only real way someone could take him down safely is by sniping or explosives but then you have the identity issue, not knowing for sure if you've got the right target and of course collateral damage in the latter case. Jones may be immune or partially immune to his influence after her experience so she alone might be the one who can actually identify him and take him down. But then how to keep him under control and locked up? Logistically impossible, really. I think she'll have to kill him.
Also, I wanted the end of Ant-Man to have Hank Pym also reveal plans for the Anthill, his prison that would shrink inmates and limit the storage space, supply requirements, and danger they present. That way even if they took Kilgrave alive, his threat would be minimal.