Detecting if another instance of the application is already running

My application needs to behave slightly differently when it loads if there is already an instance running.

I understand how to use a mutex to prevent additional instances loading, but that doesn't quite solve my problem.

For example:

Any ideas? Thankfully it doesn't need to deal with multiple user accounts or anything like that.

(C#, desktop application)

Edit: To clarify, the application doesn't need to be restricted to a single instance, just perform a slightly different start-up action if there's another instance already running. Multiple instances are fine (and expected).