A small fear of mine is that my home would be robbed, and my dog kidnapped for ransom or just hurt or killed by the savage robbers. Upon capture of them, yes, I would not only prosecute, but try and sue.
My first thought is that such lawsuits could be used in a less than responsible manner, for example in cases where the cause of death was not clearcut. If Neighbor A just wants to get back at irritating Neighbor B who may or may not have done anything to hurt little Fluffy, then it would be a total waste of the legal system. Imposing limitations on when such suits could / could not be brought would also be a large burden to the system. Of course, I don't see myself suing for "emotional distress & loss of companionship" if I lost either a human, feline, or canine companion. The money just would not make up for the pain, nor would it make me feel I'd "gotten even" so why do it?