You can request CDQs to flesh out your character in ways that cannot be done through normal game play. So if you have some "internal plot", you can advance that through a CDQ. Of course, your request has to make sense and fit within the spiritof Layonara, but the field is still pretty open in terms of what you can try to do.
Keep in mind, of course, that just because you request a CDQ does not in any way guarantee that you'll be successful, so don't look at CDQs as a free ticket to greatness.