I also have a secondary monitor hooked up, but it's a 2560x1440 monitor turned vertically. It works fine, but I forget if font scaling is applied across the board or just to each monitor.
The ideal 4k desktop monitor size is likely at least 36-38", but I'm not sure if those are economical yet or not.
At work I have a Dell 34" Ultrawide (3440x1440). Not quite 4k, but no scaling is needed, and it's a great monitor. The one downside is that it's just got the vertical height of a 2560x1440 monitor, so I kind of miss the vertical height of my home monitor at times, but I can easily have three or four files side by side in my IDE.
I also have an Ubuntu system at home hooked up to the 4k monitor at home and it works, but you'll possibly need to adjust font sizing in your apps. I haven't spent a ton of time recently with this system though.
For all of these monitors you want a video card with Displayport 1.2. You do not want to use HDMI because you will likely end up at 30hz and that is a horrid experience. HDMI 2.0 supports higher refresh rates, but having both a HDMI 2.0 port and monitor is pretty rare.
Anyways, probably too much info, but it was either do this or work on an annoying bug :)