I have a 1080 Ti and use a 1440p monitor. According to benchmark tests in some games, even that can struggle in some heavy GPU demanding titles at 4K (I have not tested it with a 4K monitor).
At this point, I would not buy a two year old GPU design and would wait for the GTX 1180 whose reviews are supposed to be out by the end of this month (Nvidia Founder's Edition). Aftermarket GPU partner cards (ASUS, EVGA, MSI, etc.) are supposed to be coming out in late August to early September.
Rumors are that it is expected to be more powerful than the 1080 Ti which currently are between $750-$800+ in the US market. The question is if the GTX 1180 will be priced less like the GTX 1080 is. We'll find out soon enough.