When in the final level, ignore the dog and go to the portal at the left. This will give you the Second Ending, the game then will take you to the Menu automatically, hit Play and you should still be at the Final Level. Now go for the dog at the right and you'll get the First Ending.

The game doesn't feature a level select, so if you go for the dog first, you'll have to play the game again to get the last achievement. Alternatively, you could make a backup of your save-game right when you're at the final level. Your savegame should be located at (Steam\steamapps\common\Alter World\Saves\savegames.sav)

You'll recognize it, because the dog will be there after you shift between worlds, also remember there's an achievement for dying at the level, so be sure to die at least once.