Well some games have crashed on me and I presumed that it was the game and the error was generated by the game. But, when it does it on the dash I get the same error so it's seems to be a system error, if that makes sense.
Have you reinstalled the firmware/rebuilt the system at all?
Reading online some systems report it as a system error when in fact it's a game causing it.
This what the support page says:
E-34878-0
CE-36329-3 An error has occurred with an application or system software.
Step 1: Close the application, install the latest system software and game update, and restart the system.
Step 2: If a PS Camera is connected to the system, remove it and reboot the system, then go to [Settings] > [User] > [Login Settings] and turn "Enable Face Recognition" OFF.
Step 3: If you have upgraded the Hard Disk Drive (HDD) in your console, please re-install the original HDD.
Step 4: If the error occurs again, back up your saved data, and then go to [Settings] > [Initialization] and choose [Initialize PS4].
Step 5: If the error continues and occurs with every application, or as soon as the application starts, please contact PlayStation Support.
It happened on mine when I first booted it up (wasn't in standby mode) and it was downloading 3/4 updates all in one go, I'm not sure if the system got confused what was going on with all the installs at once and threw a wobbly or not.
Hope it's not on it's way out, if it is then it's back to Asda for a swap.