The vast majority of the historic warhammer heads found have a pointed and a blunt end.
The horseman's pick was a weapon of Islamic origin used by cavalry during the Middle Ages in Europe. This was a type of war hammer that had a very long spike on the reverse of the hammer head. Usually this spike was slightly curved downwards, much like a miner's pickaxe. The term is sometimes used interchangeably with war hammer.
In D&D 5 however they are handled as distinct weapons. They have two lines in the Weapons table (PHB 149). This alone might not mean they are separate items, as one side is piercing, the other is bludgeoning, and the damage could differ as well. But based on the price, they are indeed separate: the War Pick costs 5 gp, the Warhammer three times as much. (This incidentally makes no sense; a point is harder to forge than a slab.)
I know D&D is not exactly medieval Europe, but I am trying to make it more logical by combining the two weapons. The user of a Warhammer could choose between bludgeoning and piercing damage as a free action on his turn.
Would this cause any balance issues?