If something someone says doesn't show up on my timeline or my notifications bar, it might as well not exist for me. Unfortunately, in a lot of cases, this happens for comments. In particuar:
* Replies to replies to my comments on other people's posts don't show notifications.
* I think replies to non-transparent shares of my posts don't show notifications.
* Comments on posts I'm interested in but didn't author don't show notifications.
I think the first two should _always_ show notifications—they're conversations that I am part of, and I want to be able to keep up with them. In the latter case, it would make more sense to be able to follow a given post and get notifications for its reshares or comments.
I think it would be cool to be able to follow a post's comments. This would be really useful for posts like this one, where the OP is asking for resources and people are leaving comments with answers.