I think it's smart to bring along a minimalist CCF pad (emphasis on minimalist). I use a 1/8" one I got from Lawson in tandem with a Thermarest NeoAir. I use a 3/16" one in cooler weather.
– At least in my area it is often impossible to 100% ensure a site is free of things that could puncture the air mattress. My understanding is that the desert dwellers find it very hard to eliminate all the spines/stickers/etc.
– The CCF pad is great for use during the day as a sit pad. In other-than-bluebird weather I always have a warm, dry, padded (lightly) place to sit and stretch out. I'd never do that with the NeoAir. I carry my CCF rolled on the outside of my pack.
– The CCF pad adds a bit of insulation/padding at night and a last-ditch backup for NeoAir failure. In a rougher situation where the weather was really threatening and the NeoAir popped, laying that 1/8" pad on top of a bushcraft bed adds a warm waterproof layer.
– A CCF pad offers many improvised options for medical care, which is great since so many of you BPL hardcores prefer 2-ounce medical kits! :)