Is there any way in C# to override a class method with an extension method?

No; an extension method never takes priority over an instance method with a suitable signature, and never participates in polymorphism (GetHashCode is a virtual method).

If the method has a different signature, then it can be done -- so in your case: no.

But otherwise you need to use inheritance to do what you are looking for.

As far as I know the answer is no, because an extension method is not an instance.It's more like an intellisense facility to me that let you call a static method using an instance of a class. I think a solution to your problem can be an interceptor that intercepts the execution of a specific method (e.g. GetHashCode()) and do something else.To use such an interceptor (like the one Castle Project provides) all objects should be instansiated using an object factory (or an IoC container in Castle) so that thier interfaces can be intercepted through a dynamic proxy generated in runtime.(Caslte also lets you intercept virtual members of classes)