My former employer allows employees to keep their work phones after they quit. When I was with my former employer, I had to install a series of corporate and company related apps so that they could monitor my phone. After leaving the company, I used the option "Erase All Content and Settings" so that my phone can start fresh. Does that mean the phone is completely disconnected from my ex-employer and I can treat it as a brand new phone?
That would really depend on how they went about doing the monitoring, on a functional level. Seeing as it was originally distributed by corporate for use in-company it's entirely possible that their apps were designed to modify the iPhone's system in a way that apps on the App Store would normally be prohibited from doing. Seeing as they were privately designed apps for use on private company property there wouldn't be much stopping them from designing the apps to inject monitoring software into the system storage, where the user cannot access or remove it, in an attempt to stop employees from simply uninstalling the apps to bypass the monitoring software.
To be completely safe I would recommend putting the iPhone in recovery mode, connecting it to a Mac with a stable (and private/secured) internet connection, and completely reinstalling the firmware & OS via iTunes' restore feature.
To put the phone into iTunes recovery mode, sometimes called "DFU mode" or "Firmware Update Mode" follow the steps below:
Warning: Make sure to disable "Find My iPhone" and delete the connected iCloud account from the device before following these steps, or it may be left unusable after the restore has finished.
1.) Remove the SIM card from the device.
2.) Press and hold both Power and Home for 30 seconds. The screen should go black around the 20 second mark, this is because the phone has shut off. The boot screen (Apple logo) will appear on screen, keep holding both the power and home button for another 10 seconds.
3.) On the 30th second the screen should turn black again. As soon as it does, let go of the power button, but keep holding down the home button for another 30 seconds.
4.) On the 60th second let go of the home button. The screen should now remain black as though the phone is powered off. If the Apple logo appears on screen try again from step 1. If the screen remains dark it means the phone is now in iTunes recovery mode.
5.) Open iTunes, plug the phone into the Mac, and a window should pop up saying that an iPhone in recovery mode has been detected, and that it has to be restored before it can be used.
6.) Click the "Update & Restore" button, and wait for the process to finish. Make sure the cable does not disconnect during any point of the restore process, if your cable is finicky grab a better one, or buy a new one.
7.) When iTunes says the process has completed, and the phone reboots: reinsert the SIM card. The phone's firmware and OS will have both been erased and reinstalled from the newest version on Apple's update servers. This means it is essentially no different from a new iPhone straight out of the box, and any spyware or monitoring programmes will definitely be gone at this point.