I just got this phone two weeks ago. It came with KitKat 14.4.A0.108 and I did it like this:
1. Backup: I don't want to sync my stuff with Google so I just exported my contacts from the contacts app to a file (.vcf) and my text messages with SMS Backup & Restore
from the Play Store. I then connected via USB and copied these backup files plus everything else I needed to my PC.
2. Enable USB Debugging and Unknown Sources on the phone. Connect the phone to your PC, it will say it was unable to install some drivers.
3. Install ADB drivers on your PC. No need for PC Companion, you can get them directly from this Sony page
Use Windows Device Manager to install these for the unknown device it shows you.
4. Get EasyRoot 12.1 from here
It may not work the first time and just hang saying "Creating VM" or "Cleaning up" or something like that. If this happens, just disconnect the phone and run it again. For me, it worked on the second attempt on my Z1 and only on the third attempt on a relative's Z1 Compact.
5. If EasyRoot is successful, install SuperSU from the Play Store.
After that, you should consider installing XZDualRecovery
so you can make Nandroid backups. Btw, none of this will unlock your boot loader, so it should be safe from a warranty point of view.