It is not to have 4 times as many things on the screen as a 1080p monitor.
It is to have a 2:1 pixel ratio (like all the apple retina displays) or somewhere in-between.
Web content, thanks partly to apple pushing high dpi displays, is now often tuned for this, showing you twice as much detail in the same space while keeping the dimensions it would have on a normal dpi display.
Read what anandtech had to say about testing a 4k monitor, and about how nice it is to look at fonts that arent just anti-aliased, but hardly have aliasing to begin with, thanks to the dpi.
I run a 1440p monitor, as it was the most pixels I could reasonably afford, (4K is just too much $) and I scale everything up so it's roughly 1080p sized. I love it for the clarity and sharpness, not for the number of things I can cram on the screen. (Although I do run my font just a little small in my text editor/ide)
There are of course downsides besides the price. Most of the 1440p monitors have poor input latency, meaning your mouse might feel a tiny bit laggy or put you at a slight disadvantage if you're a gamer, compared to lower latency 1080p monitors. That's totally ignoring whether your video card can render smoothly at that resolution. With 4K I'm not sure but I suspect it's the same or worse.