Does a 4K external video card for laptops exist?

4K what? I have never seen an external video card for a modern computer, so I'm going to lean toward "no" for any common laptop. One of the difficulties with such a thing is the laptop would need a high-bandwidth interface to connect the card to, either a proprietary interface or an external multi-lane PCI-E interface using current interface technology. Even in current desktops the graphic interface is one or more specific PCI-E slots. The inability to upgrade is one of the limitations of gaming on a laptop.

Update: Googling, I found How to make an external laptop graphics adaptor, so the answer is technically "yes".

If you want a portable gaming platform with more widespread hardware support, look into mini-ITX motherboards and cases. Some can accommodate full size video cards.

Great! Thanks for the info. Perhaps I should have said "graphics adapter" rather than "video card".

Check out 4K monitors. They're the next evolution of monitors.

Yeah, upconverting typical laptop resolutions to 4K doesn't really gain anything, and may look worse, instead of using a video card that can output native 4K resolution. 4K has 3840x2160=8,294,400 pixels. A full HD laptop has 1920x1080=2,073,600 pixels or 2,073,600/8,294,400=25% of the pixels of a 4K display. Therefore any upconversion will be fabricating 75% (3 of every 4) of the pixels displayed at 4K resolution. If the laptop has a lower resolution than HD the scaling ratio is even higher.