Hello John, and welcome to Eight Forums.
Does Outlook 2013 open if you run the .exe directly in the folder below?
(for 64-bit Windows 8)
"C:\Program Files (x86)\Microsoft Office\Office15\OUTLOOK.EXE"
(for 32-bit Windows 8)
"C:\Program Files\Microsoft Office\Office15\OUTLOOK.EXE"
If so, then you could just unpin the old icon from the taskbar, and Pin to taskbar the OUTLOOK.EXE to have a new one that works.
If not, then you might see if reinstalling Outlook 2013 to perform a repair install may help.
Hope this helps,