It shouldn't be a different user if it's a drive-image backup of the original C-drive, unless the backup/cloning tool has changed the user/computer SIDs. (That would normally only be done if the intention was to create an image which a separate PC would boot in parallel to the original which would be running on the original PC. It's not usually done for backups, only for when you want to mirror one PC to another and then run both PCs at the same time without them clashing because they both have the same identity.)
Admin Mode only elevates write operations. The main process stays running as the normal user.
Elevating everything to admin can cause other problems, and simply isn't needed in the vast majority of situations.
If you do need it, elevate the whole process, but keep in mind the issues discussed in the linked guide. It's OK to do that, as long as you understand what you're doing!
We might extend Admin Mode in the future with a way to elevate the read and write sides of things while still keeping the main process running as the normal user. That should solve your situation. But it's a very unusual situation.