* More pixels -> more characters on screen.
* More characters on screen -> easier to glance between different parts of your code , open manuals, etc.
A 4K screen has 3840x2160 pixels. You could see that as either fitting 4 regular 1080p screens in a rectangular pattern, or just over 3 regular screens in vertical orientation (which is what you were considering).
Of course, if your current HD screen is only barely comfortable to you (likely), you will end up needing to buy a screen with larger dimensions (findable) to actually be able to utilize the screen to that extent.
Alternately, you could still go for that higher pixel density; in which case you end up with more readable characters. This does alleviate eye-strain!
Given enough pixels at your disposal (and a large enough screen), you can have the luxury of determining your own favorite trade-off between pixel density and amount-of-code-on-screen. You can do so on the fly, as tasks/whims/fatigue demand.
This is why -at least for now- I'd say you can never have enough pixels.
I'm sure there's a point of diminishing returns somewhere, but I think we have a long way to go before we reach it.