This appears to be more of a user question, not a development question.
The Google Cast extension for Chrome provides two major functions:
- Chrome Mirroring - which is limited to 720p - This acts by encoding your tab to WebM/Opus and then sending to the Chromecast device.
- The Google Cast API for Chrome which allows your webpage to cast a video to a Chromecast device. Which, of course can be full 1080p. Official documentation.