A status that's a mix between "Do Not Disturb" and "Invisible"

I feel like there should be a setting or extra status option that's a mix between "Do Not Disturb" and "Invisible". For a setting, it could be something like "Receive notifications while invisible," and there could be an on/off switch. If it were a custom status, I don't know what you would call it, but it would show you as invisible, and you would also not receive notifications.