The PlayStation 4 (PS4) received the 4.00 firmware update that offers HDR color output as an option.
Most Elgato Gaming Devices Can't Use HDR Colors
Most Elgato Gaming devices cannot passthrough, capture or stream HDR video.
Thus, if you want to use the PlayStation 4 (PS4) with most Elgato Gaming devices, you must set it to output with HDR turned off.
Elgato Game Catpure 4K60 Pro Can Passthrough HDR Colors
The latest version of Elgato Game Capture 4K60 Pro can passthrough HDR colors. See this article for more details:
HDR Video is Supported with New Hardware and Software
You should also turn off HDCP.
Before you set things up, make sure to connect your PlayStation 4 directly to your TV set or display via HDMI, without using Elgato Gaming product.
Then, visit the Settings > System section of the PlayStation 4 interface, and turn off Enable HDCP.
After that, you can use your PlayStation 4 with Elgato Gaming hardware.