Disconnect your PS4 for safety, to prevent the controller from re-registering. Turn on your console, then your PS4 controller. Using a paper clip, press the RESET button in the small hole on the back of your controller. It goes off.
Turn it on BUT press HOME + SHARE simultaneously for several seconds until the light bar starts flashing. It switches to Bluetooth pairing. Quickly go to "Controller Options"> "Add Bluetooth Controller".
Just a simple pairing and... it works!