Created attachment 655688 [details]
Screenshot showing modified launcher menu (new item: "New Private Browsing
Window"; the rest are added by default on gnome-shell)

Many recent desktops support Desktop menu action items that show up when
secondary click is made on the desktop menu item. These help the user launch
the application with a different context, e.g., in the case of firefox, the
menu might have an option to launch a private browsing window directly from the
launcher and so on. I suggest we add at least the menu item for a new Private
Browsing Window to the firefox desktop menu launcher. This can be done be a
simple patch to the .desktop file:

===============================================
--- firefox.desktop.orig    2015-11-12 03:49:11.120265172 -0700
+++ firefox.desktop    2015-11-12 03:54:03.462467530 -0700
@@ -12,3 +12,9 @@

MimeType=text/html;text/xml;application/xhtml+xml;application/vnd.mozilla.xul+xml;text/mml;application/x-xpinstall;x-scheme-handler/http;x-scheme-handler/https;x-scheme-handler/ftp;
 Type=Application

+Actions=PrivateBrowsing;
+
+[Desktop Action PrivateBrowsing]
+Name=New Private Browsing Window
+Exec=firefox --private-window %u
+
===============================================

Attached is how the launcher menu looks like (on secondary mouse click) on
GNOME shell with this modification. This very modification will also work
similarly on Unity, for instance (and likely on other desktops I am not aware
of).