There is one way, that is using a meta course. That way you keep the content in a parent course and your enrollments in a child course. This can become a lot of work to maintain and you could end up with a lot of child courses. Be careful here.
I don't know of any plugin that allows that kind of flexibility, but it is a reasonable idea, set task completion dates based on the date the first time a course is entered.
You might want to go to Moodle Tracker and put that idea up as a possible new development for Moodle and a very useful tool too, I would think, for a lot of non-school, non-college/university training organization.