I bought a 6 month old unlocked S4 last week for my w1fe. It came with a Service Report from Samsung dated about 3 weeks prior stating that everything is working fine. However after only a couple of days the phone has been turning off randomly. It cannot be the battery as this is happening with the battery from another S4 and also a brand new 5200mAh Anker battery.

I haven't yet tried anything to resolve this as I thought I'd ask on DS first for any tips. My first thing will probably be to carryout a factory reset and clear any caches out to see if that helps. Beyond that I do not know what else I can do...

My own S4 is working fine and has been since I got it 15 months ago, even with the latest Android updates.

Is it turning off as in shutting itself down cleanly or instantly dying in the middle of doing other things? Does it switch itself back on afterwards? Do you have to do anything (e.g. take the battery out) to get it to switch back on?

It's dying whilst doing other things. It has to be manually switched back on. Another thing I noticed yesterday - I tried to phone it but all I got was the ring tone at my end, the other phone didn't ring. I had to call my OH to do a restart of the phone to get it making/receiving calls.