This seems like flaunting your budget more than improving your environment. Maybe I just need to get my eyes checked again, but for some reason my code just doesn't perform any better or write itself any faster with more screen real estate and I'm as productive on my laptop as I am on my big screen. 4K is not for programmers, code is for programmers.
The screen size absolutely makes a difference when you are working on a large project with lots of disparate pieces. When you jump between project-wide search, debugging, showing git diffs, and so on.
This is especially true if you do graphics work and code. When you need to move graphics assets around your filesystem and into your IDE it's nice to have the room (i.e., not so many overlapping windows) so you don't spend time digging for assets, resizing assets, running scripts over assets.
I consider myself very productive on just my laptop screen, but sometimes I postpone large architectural changes or heavy instruments based debugging sessions until I get home. There's less back-and-forth and a considerable speedup under certain workloads.