An additional product decision would be that I will assume headaches are caused if a user starts on desktop and then decides to move to mobile (Your PC must be on in order to use this mobile app!).
The real solution here is to generate a second key for (or securely transmit the original key to) the mobile device. Now the PC needs to be on to set up the second device. Once online they are completely independent.
However whatsapp currently assumes that each user has one (primary) device and only handles encryption and delivery to one device. It isn't impossible to fix (example Matrix) but it does require effort and slightly more server resources (you need to store messages for longer on average)
