It obviously shouldn't be as bright as the real thing, but it needs to be much brighter than what current displays can do. (but at the same time, I don't want everything to look that bright, hence the need for more than 8-bit color input)
My CRT does around 400 nits with ideal blacks, and even with the lights off I still wouldn't say that sunset looks bright. Something with a higher brightness (like 1000+ nits) but based on an emissive technology, maybe some kind of OLED, might produce a better effect. As it stands, the bloomed Crysis explosion gives a better "feel" of brightness to me, even though it's not actually any brighter.
Look up the Brightside DR37-P. It was an HDR LCD that was little more than a proof of concept, but it went up to 4000 nits and had individual LEDs for backlighting, so it could show true blacks as well. We need to see more stuff like that, and at more reasonable prices.