I had exactly the same problem after installing the new version 4.8.1229 on Windows Vista. Re-installing did not solve the problem, and even after installing an older version the problem persisted. I finally solved it by changing the integrity level of the program ashDisp.exe (I remembered that the same dialog window appears if you start a program with integrity level "low").
I did the following
* Started cmd.exe as system administrator
* Entered command
icacls "C:\Program Files\Alwil Software\Avast4\ashDisp.exe" /setintegritylevel medium
After logging off and on again, no dialog window appeared any more, and the icon in the system tray was displayed as well.
Hope this procedure helps in similar cases as well.