Take a standard example: hamburger patties and hamburger buns. It is understandable that if the supply of hamburger patties increases (for whatever reason), the price of these patties will decrease, thus causing an increase in the demand for hamburger buns.
Now, let's say the price of hamburger patties decreases due to a decrease in demand (maybe doctors come out saying beef causes cancer). Why would the demand of hamburger buns increase when the demand of hamburger patties decreased? Standard definition of complementary goods (that I've seen so far) implies that the price decrease of hamburger patties would cause the demand for hamburger buns to increase, but that defies common sense.
Wouldn't a more sensible definition be that goods are complementary when the demand of one good is correlated with the quantity demanded of the other?