I have a rooted sony xperia Z2 with Android 4.4.4
Is it possible to update the phone to a newer Android version and still have root permissions?
You can definitely get a newer Android version, and i suggest you Do it Regardless of the Minor Bugs...
Major Bugs are a different story, but for Minor Bugs... The added Security of even Android 5.0 will be a vast improvement over Android 4.4
To make sure that you dont already have an Official OTA ( Over The Air ) Update...
• Software Updates
And try a Update in the Settings,
If your Update Application returns that you have the Latest update, you need to do the Update by Installing a Custom Recovery & Firmware..
I looked around and found a lot of Stable options for your Model...
Official Resurrection Remix - Nougat Based
This is an official build of Resurrection remix based on Android 7 ... That is already jumping a few upgrades !
Have a look around for some Stable & Daily Use builds....