Does an analog filter with complex impulse response exist in application?
Complex impulse response makes no sense.  You put a blip into a system, and something comes out.  While complex math may be useful for modeling a system and performing operations on that model, it doesn't change the fact that any real system always has some real response.

What would it even mean for the result of a blip to be complex?  What would you expect to see on a scope if the response was complex somehow?  The real system does something when you put a blip in.  That something exists in the real world.