In article , MI
wrote:

At the end of March I bought a 27 inch iMac with 16 GB of memory, 3.4GHz
Intel Core i7 processor. It has a 3 TB fusion drive. I am running Mountain
Lion (10.8.4)

Since I bought this machine there have been occasions (about every 10 days
to 2 weeks) when the clock stops. To restart it, I have to reboot.

that sounds like hardware.

More seriously, on several occasions I have had the system freeze and I have
had difficulty getting the system to quit. It takes several tries and
several methods. Last time it happened, I got a dialogue stating that the
startup drive was full and which open programs did I want to close.

how full is your hard drive?

one thing to try is open the console app when it happens and see what
clues might be there. much of it won't make sense unless you're a
programmer, but there might be something that stands out.