Cloud recording quit too soon

Yes, the computer must stay on; unless you assigned another attendee/user as the new host before you shut your computer down, unfortunately shutting down your computer was the equivalent to ending the meeting, and therefore it stopped recording.

That is the case for a local recording that you have to wait for the file to convert. For cloud recordings if you end the meeting without stopping the recording it will end the recording. It is not the ideal way though and could have unexpected behavior like a video or audio glitch at the end. 

The benefit to a cloud recording is that the processing happens in the cloud not locally so you can immediately log off or shut down. 

I would log into your zoom.us account and check your previous meetings to see if there is a recording. You're the Admin/owner of the Zoom account can check the Meeting ID to see if there was a recording. 

you can check if it got saved to your local recordings. Go to Windows C> Users > Your Name > Documents > Zoom. This is where all recordings are stored.