The Xbox One does not have HDCP protection during gameplay, so on day one it works fine with Elgato Game Capture HD.
Xbox One: Audio
In the HDMI audio options, the Xbox One should be set to use Stereo uncompressed audio.
In Elgato Game Capture HD Settings, You can choose "Xbox One" for the Input Device, and "HDMI" for the Input. Make sure that "with Analog Audio" is NOT selected.
Xbox One: Elgato Game Capture HD Firmware
When using the Xbox One, make sure that your Elgato Game Capture HD firmware is up to date:
How do I update the firmware of Elgato Game Capture HD?
This will ensure that video passthrough will work as expected.
Xbox One: General HDMI Issues
The Xbox One has HDMI output.
Depending on what's on screen, content may be encrypted with HDCP (High-bandwidth Digital Content Protection). In that case, Elgato Game Capture HD will not be able to capture that content via HDMI.
If the output is not encrypted, like gameplay, then Elgato Game Capture HD should be able to capture content, much like it does with the Xbox 360.
The Xbox One uses HDCP protection when watching a purchased movie, or when using streaming services like Netflix. Therefore, Elgato Game Capture HD won't be able to capture that content.
For example, Netflix will not play video if Elgato Game Capture HD is attached. You can try to use Netflix full screen, or snapped to the right side of the screen, but video won't play either way. This is due to HDCP.
The Xbox One also has HDMI input.
The Xbox One can accept passthrough video from a variety of sources. HDCP protection still applies to that video.
• If the HDMI input attached to the Xbox One has HDCP protection, it will stay protected when it exits the Xbox One.
• If the HDMI input attached to the Xbox One does not have HDCP protection, it will not have HDCP when it exits the Xbox One.
Video from a cable box or satellite receiver, send over HDMI, usually has HDCP protection. That video would not play when Elgato Game Capture HD is connected, and would most likely have a black screen.