I installed Ubuntu Linux 15.10 from a USB thumb drive on my NUC6i5SYK. I had to add
nomodeset to get the LiveCD environment to boot, as the display would sleep if I didn't (as described here: Intel NUC NUC5i5RYH - Boot - only mouse pointer appears (Xubuntu 14.04, 14.10) - Ask Ubuntu and other places). I was able to successfully install Ubuntu from the LiveCD environment. Now I see the same issue with the installed version, i.e. it gets to the purple background and the display immediately sleeps. I'm connecting to a Dell U3415W via mini-DisplayPort.
I can boot the recovery mode GRUB menu item and then select resume the normal boot and it appears to work fine. Yet it consistently puts the display to sleep with a normal boot.
Things I've tried that haven't resolved the issue:
Adding i915.preliminary_hw_support=1 to GRUB boot configuration, per Early Intel Skylake Linux Users May Run Into A Silly Issue - Phoronix and other sources
Updating Intel Linux Graphics drivers to latest Ubuntu release Intel Graphics Installer for Linux* 1.2.1 | 01.org
Installing Kernel 4.3 drivers - No version of Ubuntu can be installed with any Skylake 6th generation Intel processor - Ask Ubuntu
Installing Kernel 4.4 drivers with the missing Skylake firmware file Installing 4.4 rc7 kernel yields i915 module not available