Hello
I am considering buying a 2013 or 2014 L405.
Either the Supercharged or the V8 diesel.

Are there any specific things I should look out for?
Any tips?
What options should it have?

I will be buying the car from either a dealer without long warranty or from a private person.

Thanks Range Rover 5.0 SC AB 2013 L405 Orkney grey with Santorini black roof
Land Rover Defender 2.4 TD4 2007, Java black on black
Land Rover Series III 200 tdi 1974 black on black
Swedish LR club member #1961

The Search function will help you identify problem areas.

Those years will be Euro 5 if you choose the SDV8, so consider access restrictions/charges where you might want to drive. Look at the LR website diesel v petrol article, bearing in mind they want to keep selling diesels. 2013 SDV8s may not have the 'turbo drain mod' (search for implications). At that age SDV8s that have done lots of short journeys might start showing problems, so research driving history if possible. Some 5.0SCs of that era might have issues with chain rattle (again search).

Options:
Vogues would not have had adaptive cruise control as standard, VSE and AB do. There are advantages and disadvantages with each, depending on where/how you drive. Eg adaptive stops working if the sensor on the front of the car becomes obscured by snow.
Pano roof seems desirable, but solid metal roofs very rare anyway.
The wider pillars needed for structural integrity in aluminium make blind spot monitoring desirable. Also reverse traffic detection if available and relevant to where you will be driving.
The VSE and AB semi-aniline seats are more comfortable than those in the Vogue, and the winged headrests are nicer too. Again depending on where you live/drive, soft door close is useful.
The standard audio is OK for most things, but even at VSE/AB level drones/buzzes a bit in some classical music. The top audio option is attractive if that matters to you, expensive to select new so relatively rare.
Avoid 22" wheels if planning to drive off road. I have 21s for summer and 20s for winter tyres. Pirelli Scorpion Verde have low rolling resistance and so good for fuel economy/noise, but can 'tramline' a bit.
The farmer's hitch towbar is cheaper and simpler with higher nose weight rating than the deployable, less to go wrong and quick and easy to attach/remove. Only Range Rovers since 1988

About 15k between a diesel and a petrol strangely  _____________________________________________________
Zambezi silver Vogue Se unique

You have a 2011 Autobiography... so I would look for an Autobiography spec car. It has most things you could want.

If mileage isn’t high, I think S/C. Euro V diesels will suffer worse depreciation in my view.

The 1700w Meridian is awesome. Doesn’t add much to 2nd hand value but seeking it out will reduce choice. I also have the heads up display which I like but is quite rare so wouldn’t be a must have. Current: L405 SVAutobiography Dynamic 2018, Defender D240 HSE 2020, Defender 110 Adventure 2015, Defender 90 XS 2012, Lightweight Ser III 1975
Waiting for: L460
Previous: L405 SDV8 Autobiography (2016 x 2, 2015 x1), 110 V8 1989, Discovery td5 ES Series II 2000

Thanks for all the info.
I have found a 2013 RR 5.0 Supercharger, Autobiography that I intend to have a closer look at.
It has 48.000 miles on the clock.
3 owners, full service history.
The usual options incl massage seats, panorama roof, 22" wheels, Orkney grey with black roof.

It is unfortunately located 400 miles away

Listed at 49.000 GBP

I will let you know what happens  Range Rover 5.0 SC AB 2013 L405 Orkney grey with Santorini black roof
Land Rover Defender 2.4 TD4 2007, Java black on black
Land Rover Series III 200 tdi 1974 black on black
Swedish LR club member #1961

I bought it!
If it was good or not - only time will tell.

Thanks for all advice.