[Xperia Z1 Q&A, Help & Troubleshooting] x016x:
Rooting on the newest version of Android 4.3.3 isn't impossible on the Xperia Z1, it just takes you more steps than normal to actually get full root access. I have a locked bootloader as well but I was able to get a rooted version of Android 4.3.3, specifically the iHackers 2.0 ROM running Android 4.3.3 with full root access. Here's how I got full root access on the iHackers 2.0 ROM:

1) Download the .ftf file named C6903 14.1.G.1.534 Global 1277-2880_R4A from the compiled FTF thread at this link: http://xdaforums.com/showthread.php?t=2469191

This is a version of the original Stock Rom that the VROOT program will actually work on when trying to ROOT your phone.

2) Flash the ROM you downloaded using Flashtool (steps on how you can go about doing that should be in the Xperia Z1 General forum.) and now, you're ready to root your phone with it.

3) Download, install and run VROOT to root your phone. Again, more information on how to properly root your phone using VROOT can be found in the Xperia Z1 General forum. For me, I just turned on USB Debugging on my Z1, connected my phone to my computer while VROOT was running and clicked the green button until I got root access. It's extremely easy.

4) With Root Access, now you can download and install Dual Recovery for the Xperia Z1 on your phone.

5) Download the latest iHackers 2.0 ROM from the Xperia Z1 Android Development thread and copy it to your phone. This is hands down, the best ROM for the Xperia Z1 right now in terms of battery life. With the latest 4.3 update, I'm able to get over 2 Days worth of battery life on one charge. Battery life is much better on the new iHackers 2.0 ROM running Android 4.3.

6) Download the flashable Locked Dual Recovery for the Xperia Z1 here. http://nut.xperia-files.com/

The file is called XZ-lockeddualrecovery2.7.63-BETA.flashable.zip (8.48MB), download it and copy it to your phone, preferably the same place you copied the iHackers 2.0 ROM. To make things easier for yourself, create a new folder on your phone or SD Card in your phone and name it iHackers. Copy both the iHackers 2.0 ROM and the flashable Locked Dual Recovery.zip file into this folder.

7) With both files now copied to a folder on your phone, you're now ready to go into recovery to flash the iHackers 2.0 ROM. Boot into recovery and once you're there, do a Wipe Data/Factory Reset before Installing the ROM.

The rest of these steps are now taken directly from the iHackers 2.0 download thread on the Xperia Z1 Android Development forum.

8) Install the ROM and THEN Dual Recovery. Don't make the same mistake I did, once the installer is finished, DON'T REBOOT THE DEVICE. Uncheck the "Reboot Device" box and go back to recovery to install the flashable Dual Recovery.zip file that's in the iHackers 2.0 folder.

9) Switch Off Your Phone (Do Not Reboot). Select No to Disable Flash Recovery Confirmation.

10) Install Kernel.ftf for C6903 (.290) or Kernel.ftf for C6902 (.290) with Flashtool to get compatible Kernel and Baseband. To download the kernel.ftf file, go to the original iHackers 2.0 thread to download and flash the kernel using Flashtool. You should be familiar with Flashtool at this point since if you make it to this step, you've already used Flashtool once to flash the .534 baseband version of Android.

11) Reboot and Enjoy. Now you're running the latest version of Android 4.3 on your Xperia Z1 on a Locked Bootloader WITH full Root Access. Sure, it's not the stock version but the iHackers 2.0 version of Android 4.3 is just as good, if not better. This will certainly do until there's a better way to simply root the newest Stock Version of Android 4.3.

I've been looking into unlocking my bootloader to install custom kernels on my Z1 but the battery life and performance on my Z1 with a locked bootloader is already pretty impressive. I can't imagine how good it could be with a custom kernel.