That's tricky... and I'm simply guessing here. So you shut off the car, open the door, and they remain on? This is the headlight and not the fog lights, correct? When you test drove, did the seller hook the battery up just before? On mine, both the stalk and the BCU can activate the headlight relay.
You could try pulling the negative battery post, wait for 2 minutes, then reconnect. This could reset the BCU.
I'm not sure on the '13, but on the '15 there are 2 ways to turn off the lights. One is to twist the stalk to off. The other is a menu setting, where you can have them go on/off with the ignition. If equipped, you might use that option and see if they go off. The stalk should override the BCU. The BCU is simply another way to shut off the headlights.
Start the car then turn off the stalk switch and see if they go out. If I turn on the headlights after shutting down, lights will stay on and don't time out.
Next try the auto-shut off setting, start the car, then power off. If they still remain on, I'd check the fuse panel. I believe yours has a fuse and a relay in that box. I would start by swapping out the relay (after removing the negative battery cable). It is possible the relay contacts could be stuck. The stalk switch and BCU simply turn on this relay, which then turns on the lights.
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