When I buy things in the Diablo 3 auction house, it says I completed the purchase (gold purchases, not cash) but when I go to my stash in game, it's not there. Or in my inventory. I only am using 1 hero right now also.
You have to send items purchased from the auction house to your stash explicitly. This extra step is so you can purchase things you don't actually have space in your stash for, it lets you queue up to 50 things in the auction house "outside" of the game.
You do so from the left side of the Completed tab.
Select the item purchased, then hit "Send To Stash".
Pictured here is sending gold from the sale of an item, but the same system applies to buying items.
It doesn't really make sense that you have to send gold to the stash IMO, but whatever.