^^Definitely true for purple. 
I pretty much love all those doors in that link! I do love a yellow door on the right house. Or pink. Or any color!
I’m a big fan of Benjamin Moore Essex Green for front door and shutters. It’s much deeper and richer than forest green and will be a lovely contrast to your brick without being jarring and garish.
I don’t think the door has to be same color as the shutters.
Another “No” to the same color as shutters.