OSX - Only show apps with open windows in the dock?

I like to keep my dock tidy with almost only the applications I'm currently using shown. As such, I tend to completely quit out of apps so that they don't clutter my dock. However, there are apps which would be advantageous to allow to update in the background (e.g. mail). Is there a way to allow an app to remain open, but hide the icon when there are no open windows? I've seen ways to completely hide the icon at all times, but I would prefer that the icon was there when there are active windows open. Any clean way of doing this? Thank you much.

Solution 1:

Hide the Dock, then use Contexts for Mac to switch between active windows. Contexts has an option to show only the application with active windows regardless it’s running in background or not. You can read my review here.