I have two scripts located in C:\\setup: script.ps1 and script1.ps1.
I want to be able to run the script1.ps1 from withing script.ps1 as another user and with elevated p
Since you're concerned about the new session window closing, I'm guessing you want command line output.
Start-Process is working as intended. It will run the script passed in through -ArgumentList and exit the session. This means it will not hold to display command line output - the session will terminate immediately after the process completes.
If you want a persistent session, use New-PSSession. Otherwise, you could export the data you're gathering to a file.