After searching the net, this is a common problem with the ASUS ROG PG348Q.
However, I think I found a solution.
When the ASUS ROG PG348Q will not turn back on after being powered off, I found I had to do two things.
- Unplug the monitor from the power supply; AND
- Unplug the monitor from the display port in the computer.
It seems me that somehow the computer display port (in my case the MacPro) is sending some signal which is blocking the power-on function; maybe it is part of a "sleep mode" handshake between the computer and the monitor, which is buggy, I'm not sure.
I wish this great monitor just had a simple, mechanical ON/OFF switch; but I guess those days are long gone