I was thinking along the lines of a synthetic filament material here, not tungsten which is already at its limits. If we could find something which conducts electricity plus remains solid at as high a temperature as possible then incandescents may still yet have a future. As for color temperature preference, the beauty of a 5000K or even 6000K filament material is that you don't have to run it near its limits if you don't want to. You can run it at 3700K, 4000K, anything up to its maximum rated temperature, depending upon your preference. The only major problem I see with much higher temperature filaments would be the need to block UV, perhaps via a UV reflective coating which can also reduce power consumption similar to the way present IR coatings work.
I don't think we'll ever see a practical solid material able to physically reach temperatures higher than tungsten or carbon (around 3700K). It is certainly possible to reach apparent color temperatures higher than that using incandescent technology by using things like filters and reflective coatings -- I know there has been some work done with IR reflective coating, and I'm sure there's a lot of potential to go there. However, I expect that LEDs will both drop in cost, and improve in capability to the point where it will actually be cheaper to go solid state, than with some exotic next-generation incandescent, for effectively the same thing.
In the shorter term, I'd like to see some more options become available in small-scale HID lights and ballasts (read: the 10-50 watt range suitable for flashlights).
Although my sweet spot for general lighting seems to be in the 4500K to 5500K range, I find CCTs as low as 3500K or as high as 6500K acceptable.
Indoors, for ambient lighting, I prefer 3500K. I actually find I can see better at that color temp and actually get away with less light -- For example, I find that a 9W 3500K CFL (500 lumens) is a suitable replacement for a 60W incan, or 14W 2700K CFL (850 lumens). For task lighting, I prefer 5000K.
Outdoors, my preference is HID at about 4000K (true for flashlights, street lights, and auto headlights -- provided glare is properly managed). The closer to that, the better -- the LEDs that I prefer are Cree WH bin, around 5000K, augmented with a small amount of red light. The incans I prefer are the ones being driven the hardest, 3200K+.