Share your experience!

I am facing the same issue. It works only when the phone is connected to a power source. Even with 100% charge, once you disconnect the charger, it shuts down automatically. This has happened after the Lollipop update only.
Now this is where it gets super weird.......
The small pin on the micro USB port which connects to the charger broke on my phone. I had gone to the Authorized Service Center of Sony and they simply refused to touch my phone stating that since its a Z series phone which is water and dust resistant, the company does not allow them to touch those phones. Spoke on their online customer support who echoes the same thing and suggested me to send the phone to the company itself via the service center. He told me if I decide to do that, i have to pay a handling charge irrespective of whether my phone gets repaired or not.
So i decided to find another solution. I got myself a Sony Dock Charger for Xperia Z which was specifically launched for users like me whose pins have broken. Now that Dock charger does not connect to my laptop no matter what I do. As in my laptop doesn't recognize the phone when its connected.
So, PC companion is also not possible for me.
The dock is only to charge the phone, so it will never work with PCC, with that said, there isn't much to suggest here, I'm sorry.
So that means the time has come to throw my phone into a dust bin.
Huge mistake I did by buying A Sony Phone. Never buying again.