(Unfortunately you don't state what version of Adapt you're using - so I'm going to assume it's v5. You should be able to do the same thing in v4 and earlier it's just the class names will be slightly different)
is-complete class is added to the
boxmenu-item div when that menu item is completed, you could use that to have a different background colour for completed items:
unfortunately doing the same thing for an incomplete (as in 'started but not completed') menu item isn't so easy because there's nothing in Adapt to handle this, there's is-visited but that's not quite the same thing. See this ticket.