Linux

Good luck with this approach on a server.

Indeed, obviously I'm talking about desktops here.

If by ‘suspend’ you mean that the process will just halt, then: Which processes? All of them? Good luck displaying a message then.

You could use some kind of heuristic to suspend ones using the most memory/CPU. Or just suspend them all. Obviously you would exclude the processes needed to display the message.

If by ‘suspend’ you mean moving the memory to disk

No I meant just pausing their execution. I'm pretty sure ctrl-alt-del does something like this on Windows.