Hi Mintfresh,
I had by big box 2000 hooked up with a Cocolino adapter for the mouse and a Lyra for the keyboard. Both devices were from Amigakit. Then I had a
Microway FlickerFixer AGA 2000 card allowing me to connect my Amiga into a modern monitor. I use a Samsung flat screen monitor.
I had the Amiga and three PCs all share the same monitor, keyboard and mouse through a Data Transfer Switch: An "ab box".
It worked just fine. So yes, it's possible.
If you have a hard time finding a video card for your 1200, perhaps something like
this would help.
Fester