If after this you still get the compatibility mode warning then it means windows set the compatibility mode in a registry key that can't be toggled off with the UI. In that case the only way to stop the game or steam running in compatibility mode is to delete the corresponding registry entries. To remove the steam and coh2 compatibility mode registry keys manually do the following:

1. Completely exit COH2 and Steam
2. Run regedit (start menu -> run -> regedit )
3. Back up your registry just in case you delete or change the wrong key (use export)
4. Check the following locations for any entries mentioning RelicCOH2.exe or Steam.exe and delete the keys inside this locations that mention these programs (in the right panel of regedit right click on the line with the .exe and select delete )
HKEY_CURRENT_USER\Software\Microsoft\Windows NT\CurrentVersion\AppCompatFlags\Layers
HKEY_LOCAL_MACHINE\Software\Microsoft\Windows NT\CurrentVersion\AppCompatFlags\Layers
HKEY_CURRENT_USER\Software\Microsoft\Windows NT\CurrentVersion\AppCompatFlags\Compatibility Assistant\Persisted
5. Exit regedit
6. Start Steam
7. Run COH2

Thanks to a guy on Steam for providing this info.