I don't have my xperia z with me to check firmware number but I did update to 4.3 from 4.1.2 (what it came with fresh out of the box)
Towel root worked.
Just go to towelroot.com (I believe that's the link) download apk by clicking the picture.
Click "make it rain" . It says you're phone will reboot (mine did not). Instead of rebooting I instantly downloaded root checker app to verify.
Congratulations you have root access was the result so I downloaded my su app of choice. Updated binary...
I finally restart the phone.
Everything worked great. Also installed busy box (get pro version....select smart install) works flawlessly
Easiest root to date for me (my Samsung galaxy light required my pc, odin and a tar file)