Q: Mac mini (Late 2012) Video Problems

That looks like a graphics problem.

Since your Mac uses system memory as video memory do you have the original memory installed?

You can try taking out one stick of memory and see if problem. If problem try reinserting that stick and taking out the other stick.

You can also try the following to help rule out a software proboem

- Try resetting the SMC and NVRAM/PRAM

Intel-based Macs: Resetting the System Management Controller (SMC)

https://support.apple.com/en-us/HT201295

About NVRAM and PRAM 

https://support.apple.com/en-us/HT204063

- Try starting in Safe Mode

OS X: What is Safe Boot, Safe Mode?

https://support.apple.com/en-us/HT201262

- Boot to Recovery and repair the startup disk

OS X: About OS X Recovery - Apple Support

https://support.apple.com/en-us/HT201314

- If it is repairable reinstall the OSX

How to reinstall OS X on your Mac - Apple Support

https://support.apple.com/en-us/HT204904

- If you do not have a backup use disk utility to restore the internal disk to an external disk so you can try to recover data.

If still problem that would indicate a hardware problem. You can try running Apple hardware Test

Using Apple Hardware Test - Apple Support