I can't play movies on my computer from Amazon in HD. TV shows play in 1080p just fine. I've tried using Chrome, Firefox, even Edge and IE11. I contacted Amazon support and all they did was have me clear my browser cache and then offer a refund.
It's not always an obvious question to some and isn't meant to be condescending but, is your internet connection fast enough to support a 1080p stream?
When you're using Internet Explorer as your web browser you'll need an up to date version of Silverlight installed.
Otherwise, make sure you're using the most up to date version of each browser.
Finally, Amazon states that for HD support to work, your system must conform to and have HDCP enabled, otherwise they will provide you the SD version as a fall back.
The easiest way I know of to check the HDCP status of your configuration is to download and run Cyberlink's Ultra HD Blu-ray Adviser tool that checks for Blu-ray playback compatibility. It works for BR, BR3D, and UltraHD. It tells you the details of what is and isn't ready in your system for those tasks, but the key for your situation is, it will tell you your HDCP status.
Yeah I've run the cyberlink tool and my pc is hdcp compliant, Radeon RX 480 connected to an acer 1440p 144hz displays over Displayport. My internet is 200mbps down/10mbps up, and plays 1080p amazon tv shows and 4k netflix without any issue. I used IE just for the sake of testing all the browsers on my computer. Both Firefox and Chrome are up to date.