Great idea but I think more of us cyclists could be riding and anticipating 'if that car door opens can I stop in time?'. We can't rely on people in cars doing the right thing. Many of us cycle too fast to be able to react in time.
It's impossible to dodge a door open by an uncareful driver. It happened to me, I was cycling really slow, I was watching that door because I suspected the driver was still inside: When the driver slammed her door open, it just hit my shoulder, there was nothing I could do, it happened too fast. Since that day, I just constantly ring my bell when bike by car doors, which must be very annoying for the neighbours. But ey, safety first.
One more reason it's difficult to dodge an opening door is, swaying out of the cycle lane may lead us to hit a car we haven't seen.