From the Amazon Prime Video website:
Ultra High Definition
Ultra High Definition (UHD) offers a "4K" picture resolution that is four times greater than HD. At this time, Ultra HD playback is supported on the following Amazon Instant Video compatible devices:
Samsung Ultra HD TVs (2014 or later models)
Sony Ultra HD TVs (2014 or later models)
LG Ultra HD TVs (2014 or later models)
Vizio Ultra HD TVs (2014 or later models)
You need a high-speed Internet connection of at least 15 Mbps to watch videos in UHD. You may also need to upgrade the firmware for your compatible Ultra HD TV to the latest version. You'll know you have the latest version if you see an Ultra HD icon on the Amazon Instant Video Home screen. For more information, go to About Updating Your Device Firmware.
Tip: If you're using a Samsung Ultra HD TV, you might also need to update the Amazon Instant Video widget on your device to access UHD content. A blue "arrow" displays on the Amazon Instant Video icon if an update for the widget is available. To update, select the Amazon Instant Video icon, press and hold OK, and then select Update Apps.
To browse movies and TV shows available in UHD, select the Watch Ultra HD category on the Amazon Instant Video Home screen. Titles offered in Ultra HD feature a "UHD" or "Ultra HD" label and purchase options in the video details.
Note: We’re continuing to expand our selection of Ultra HD movies and TV shows, but not all Amazon Instant Video titles are currently available in this format. We're constantly working to establish and improve our agreements with content owners so we can offer more titles in UHD.