Two world_ clock.exe running at the same time - this is abnormal situation and is not supposed to be happening. world_clock.exe has code in it to exit if it detects another instance running (or something like this). One thing that may not be taken care of if something starts both processes at approximately the same time, then each of them may have a bit of a problem determining if another instance is already initialized.
I suggest trying a few things.
1) After killing secondary world_clock.exe, exit the first instance normally via right click on clock - Exit, this should terminate the program in normal way. At this time you should not see world_clock.exe in the list of running processes. Then start World Clock from the Start button in Windows - normal result is one world_clock.exe running. If you see two, something is very wrong.
2) Check the list of installed programs in Control Panel to see if you maybe have multiple versions of Anuko World Clock, or ZoneTick (former name) installed in different directories perhaps. Make sure you have only one World Clock program installed.
3) Try uninstalling and rebooting. At this point there should be no world_clock.exe running. If you see it running, see item #2. If everything is good after uninstall, install it again and reboot, normal result is one instance of world_clock.exe running.
4) Perhaps check your system for malware with antivirus, something may be playing up here. Check exe for certificates, all World Clock components (exe and dlls) are signed with Anuko Certificate.