Develop for 4K resolution on desktop, or let browsers handle the scaling?

Use media queries to accommodate the extra use-case. Even if browsers do eventually wise up and start to treat 1px differently for different desktop screens, this could take some time and meanwhile your site looks crap on those 4K screens.

For a guide on how to handle 4K you can check out this article, which is helpful: http://www.infiniteproduction.eu/blog/building-4k-uhd-website-3840-2160

The basic gist is that you want to scale up your layout, make your images vectors where possible so they can also be scaled up, and scale up your font sizes.

When/if browsers do eventually gain support for 4K resolution screens, your site will still render nicely, so there's really nothing to lose.

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Link to an archived version of the above.