For me, completing a Pokemon game would mean defeating all trainers (path, gym, elite, champions, including the 2nd post-game rematches), collecting all the ground items (including the hidden items), and catching all available pokemon in that particular version (grass, water, cave, legendary, swarm, trees, safari zone, not including the other choices for starters and multiple choice bonus Pokemon). I haven't touched my HeartGold game in a little while, but I was sure I got almost all hidden items, but I'm not even close to catching all the Pokemon available in that version).