Amazon Music does not have a desktop client for Linux. The web application works fine but is restricted to SD only. After rooting about a bit, I found that using bottles gets you an easy working install of the Amazon Music Windows application on Linux that allows you to play HD and UltraHD tracks.
Download bottles from here, then create a new bottle for Amazon Music. You now need the Amazon Music Installer executable.
Annoyingly, I was not able to download this on Linux using any browser – they all refused to get the content and I was too impatient to figure out how to get it through Curl. I downloaded it from a Windows VM instead and put it in a shared location. Now run this executable in your bottle and then launch the installed application from the bottle to enjoy Amazon Music.
[Edit] I figured out that this is due to user agent sniffing by Amazon Music. On Edge, use Developer Tools, Network Tab, Change User Agent to Custom and pick the Windows Edge User Agent configuration to download the executable on Linux.
You can tweak settings like font DPI to make the display better. I haven’t played around enough to get the display looking better other than increasing DPI to 120 but it is quite usable out of the box after installing DXVK and allfonts dependencies.
As far as I can tell, this does not let Amazon Music integrate with my KDE desktop as a media player (media keys do not work, desktop media controls do not apply to the app etc). For now, I’m ignoring that problem as I am just happy to be able to listen to higher quality sound on my main home working environment.