I can help you. I just did it a couple of hours ago and the step actually is quite easy but since it takes many steps, it might spend you one to two hours to learn if you aren't knowledgeable about flashing stuffs. I was on unlocked bootloader with .757 rooted but that should be no different at all if you don't flash custom kernel(I didn't do that too because currently no .108 custom kernel for us).
Be sure to backup your whole existing rom in recovery first and copy it to your computer
Use Titanium backup to backup all user apps and app data and copy it to your computer
Make sure your contact list in phone book is backed up by Google. If you aren't sure about that, remove your account in PlayStore and setup it again. You will see it asks you permission to backup contact list to Google via Internet in last step(default is yes).
Install the latest ver of Sony pc companion in your windows and reboot. It will install the correct z1c driver into your system. Make sure you used a good quality usb cable because if not, that can cause serious problem during flashing firmware and brick your phone in worse case. You can also try to install Xperia driver set in the link above if your windows just can't install Sony pc companion successfully but I think PCC should be better.
I used the rom in this thread. But I was having problem to download it with firefox, everytime it was stopped during in the middle progress, it might be conflicted with some of my add-on. At last, I use Chrome to download it and completely no problem at all.
Download the latest flashtool from here and put the D5503_14.4.A.0.108_Generic_UK_1279-4053.ftf under /firmwares
Enable USB debugging mode and allow install app from unknown sources in your phone. When you first time to enable usb debugging and connected the phone to windows, there will be a screen to ask you accept and remember on your phone screen. Please immediately to click it.
Run flashtool(FlashTool for 32bit win, FlashTool64 for 64bit win). Click on the first icon(power icon) and select flash mode. It will open up a windows to let you select .ftf, pick the Generic_UK_1279-4053. Uncheck the DATA if you want to keep the existing system settings and apps and apps data. But it would be very easy to cause problem like battery draining and unstable after updated because the system settings in .757 and .108 is different. I would highly recommend you clean the DATA here and restore the apps and apps data from Titanium backup later. Please note that you have to quit PC Companion before you run flashtool, otherwise they may have chance to conflict.
Follow the guide here to enter flash mode. First power off your phone, second unplug the usb cable out of the phone, third press and remember to keep holding down the volume down key then connect the usb cable to your phone again. After a few seconds, my windows7 showed installing device driver at the right corner and wait it for a couple of mins it will be done and tell you flash drive installed. I still keep hold on the volume down at this moment but I'm not sure if it really needed. The guide in the link said there will be a Green LED light on xperia z but my z1c just didn't show me any LED light.
I remember the flashtool was aborted because I was unpluged the usb cable out(though I reconnected it later on). I was once again selected the .ftf and this time it immediately detected my phone was in flash mode and started to flash the firmware to my phone. I released the volume down key at this time. Remember to close all software in your windows, don't listen music or play games please, don't do anything and don't let any people near your computer now(if your computer health is bad and always easy to hang, you should give up now until fixed the problem or buy a new computer). If your windows is unstable and got blue screen, your phone will brick. Just wait it to finish, not too long, I think 10 to 15mins.
Your should flashed .108 successfully now and unplug the usb cable, press on the Power to let the phone boot into OS.
Repeat step seven again.
Use the EasyRoot tool v11 here and run it in windows. After rooted then reboot first and go to playstore to install SuperSU(by Chainfire ver) and reboot again. Please note that sometimes EasyRoot tool v11 will get stuck during in the middle processing for some unknown reason, don't worry, just press Ctrl-C to cancel it and run it again.
I've used Z1C-lockeddualrecovery2.7.150-BETA.installer.zip and please download it from the sendspace link above and run it in windows(it also worked perfectly on my unlocked z1c). In the command prompt window, select "1" - SuperSU and it will install custom recovery to your z1c. It will also install a NDR Utils app and you can select which recovery(cwm or twrp) you want to use in that app, default is TWRP. Please note that there are some new ver of Z1C-lockeddualrecovery but some may buggy and I don't have time to test them all.
Now you've done but if you want to deodex the rom to play more tweaking stuffs/mods. There are more steps to do.
Install the latest ver of Java runtime for windows and reboot.
Use kamome tool v6a and run it in windows. Follow this steps: 1>2>3>5>4>6 and at last, copy the deodex_install.zip to your phone storage. Reboot into recovery and flash the deodex_install.zip, clean cache and dalvik cache then reboot. Please note that I think kamome tool v6a will also install a SuperUser into your phone which conflict with SuperSU. Don't worry, click on SuperSU and it will tell you SuperUser detected and ask you to uninstall it and click Yes.
Restore all apps and apps data from Titanium backup but if you are originally come from an odexed rom, don't restore those odex apps into a deodexed system, you have to install all of them via PlayStore manually first then only restore the app data in Titanium backup thereafter.
You have finished pretty much all things now. It will take you many steps to go but all are easy work.