This is going to be controversial. And for those developers who's managers read this post and decline their, likely legitimate, request for a 4K screen, I apologise. But, not all developers need a 4K screen for their work. I don't and, trust me, I have given it some serious consideration.
We've all heard of Moore's Law, Murphy's Law, and all kinds of other ones. Years ago, I coined "Bell's Law", which is that the number of monitors you have is always at least one fewer than you need at any given point in time.
As I sit here, I've got this blog post open on one screen as I type; some ice hockey on another screen for entertainment; some notes for this post and WhatsApp open on another screen. Whilst all that's happening, my phone is going off for emails and Twitter. I need a screen for those.
When I'm at work, I have my emails and other communication open on one screen. A screen for my code. A screen for testing what I'm writing. I'm out of screens. I need at least 2 more: 1 one the database I'm working with so I can check things over, and one for general research, documentation, or just a reminder of the spec.
Those scenarios are what drew me to a 4K screen. I could, in theory, have 4 x 1920x1080 screens in one. If I got 2 of those, I could have 8 x 1920x1080 screens across those two monitors; plus the one on my laptop. I'm sure I could fill all of that space quite comfortably and still need more screens, but it very quickly falls down.
My current setup has a 17" laptop screen, as well as a 23" and 24" monitors to the side. The problem, therefore, is this: 4K monitors at the lower end of the price spectrum are around 27", that would give an approximate diagonal of 13.5" for each of those monitors. That's pretty small. My personal laptop is a 13" device, and the thought of trying to work with 9 of those screens all day frustrates me. I choose to have roughly 24" monitors as external ones because they are easy to read text on without straining my eyes.
To get the similar size windows on each of the 1920x1080 panels, I'd need to be surrounded by 2x 40" screens. That's bigger than some people have as their primary TV. Whilst I'm sure it would look fairly cool to the casual observer, the thought of that amount of light being bestowed on me all day makes my eyes hurt. On top of that, I only sit around 27" from my screens, so they would be looming extremely close to me.
No, I'd sooner have one or two additional 24" screens running at full HD to spread my normal workload through (my laptop won't power that many though). That's not to say developers don't need 4K monitors.
When you need a 4K monitor
There are cases where developers legitimately need a 4K screen. The main one here is someone working in games development for those with 4K screens. You'd need to see how things are working together. Anyone who is doing design and development within the same role; a 4K monitor is likely to be handy there.
Basically, anyone who is doing a lot of work with graphics is likely to benefit from a higher resolution screen. I just don't think it's something I need.
A final note to those managers who are using this post to deny a developer a 4K screen. If they feel they need one for their job and to be productive, then get them it. They likely know better than you how to make them productive, and the reasons why. The boost in morale and productivity alone will be more than the £300 it costs for a monitor!