Slukse is correct -- it depends on what you mean by "alongside."
Obviously you can't compile C# code into a SWF -- that much we know. But you can certainly embed (as davr suggests) an ActiveX control into a .NET Forms app, load the SWF into that control, and use the ExternalInterface API to bridge the C# code running in the desktop app and the ActionScript code running in the SWF. If by alongside you mean writing server-side C# code to render data consumable by the Flex app, then of course, you can do that, too -- there's plenty of documentation out there covering how to connect a Flash or Flex (or AIR, by extension) app to Web Services of a variety of flavors, including the relatively new-ish WCF stuff.
Another way to go would be to run a standalone C# desktop app (e.g., a service, console app or the like) and have that app listening over a certain local port -- then have your SWF talk to that app using the Socket classes. I'm doing something like this now (with Java, not C#, but the idea's the same) for a personal project.