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I thought the safe zones were horizontal including corners...
I'll be running the socket horizontally from the socket, and the new socket will be at the same height.
Soul-
I was questioning myself as I wrote it. I have done it before but didn't think it was allowed. Maybe someone else can confirm. Further searching reveals it's muddy waters.
The underfloor route would often be preferred as corners tend to have a bit more going on structurally but that depends on the age and construction and I think you're in a new build?
I see now why you are thinking trunking would ensure the wires stay on the horizontal. In my experience trunking is not often used in walls. There's a type of capping which is useful in solid walls if you are plastering over the chase.
Airhead
If you're chasing the wall you are going to be flouting the rules a bit because cables should only be running horizontally or vertically from visible outlets on the wall. I have done this going round corners myself but strictly speaking it's not allowed. There is an area at the edges of walls where wires can be run without accessories visible on the surface but that doesn't apply in your case.
Normally you would go down, under the floor and back up to get to where you want the socket. Depending on the subfloor type you can create access hatches or lift boards and pull the wires through.
http://www.ultimatehandyman.co.uk/how-to/electrics/permitted-wiring-zones