Odd that the most obvious hypothesis wasn't given: the crow was probably using its alarm call to warn other crows. Crows are social animals, who care about the fates of their "friends and acquaintances" — so they would do that.

But also, on a tangent, there is a bird that does this kind of non-conspecific alarm calling the time as part of its food-gathering strategy: the African fork-tailed drongo.

The drongo gives true alarm calls to food-rival species nearby, to tell them when it has spotted a mutual predator. This leads to these food-rival species coming to rely on these signals. But then, every once in a while, it gives a false alarm, to get the food-rivals to run away for a bit, so it can nab the bugs/berries/etc that the rival would have been eating.