Why not just use your phone then. Many android phones can launch into a desktop environment
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Do you also need to have power for the PC from USB-C, or can you live with wall power for that?
I went looking for PSUs that used USB-C a while back, and that's kind of limited.
If that isn't an issue, then I think all you need to look for is USB-C DisplayPort Alt mode for video. Everything else should just work if you have a USB-C connection, assuming that you've got the monitor side of things handled.
EDIT: If you have to have power via that USB-C connection, this is two years old, but:
https://www.reddit.com/r/MiniPCs/comments/1f9m7lv/mini_pc_with_usbc_power_and_display/
The only ones that I know that have both power delivery and display out using the same C cable are Minisforum EM680 and EM780.
EDIT2: Sorry, that's PD as a host rather than as a device on the Reddit thread.
Instead of swapping the entire dock, I have a USB switch for mouse and keyboard and then an HDMI switch for the monitors. Also let's you mix and match monitors easily
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