Mixing recipes wouldn't give you the same flavors. They need to develop separately to give them the distinct IPA/Stout flavor each one has. If you want to brew just one beer and depending what flavor profiles you like in each, you should be able to find a recipe that matches what you're looking...
I like to pitch two different yeast strains at the same time. Now while a Belgian with the European ale may not be my first choice, it may turn out great. Won't be in style, but could be an experimental brew.