I did a google search for split pea soup and ran across the recipe for Andersen's, which is a restaurant in California, and was very happy with the result.
What you need:
2 quarts water
2 cups split peas (14-16oz bag)
1 branch of celery, chopped
1 large carrot, chopped
1 small onion, chopped
1 clove garlic, minced
1/4 tsp thyme
pinch of cayenne
1 bay leaf
1 tsp salt
1/4 tsp black pepper
I put all of this into a large pot and turned it up to high.
Bring it to a hard boil for 20 minutes. It will start to look frothy like this:
Reduce heat and simmer for 20-30 minutes, or until peas are soft and start falling apart. Once the peas are starting to get a bit mushy, remove the bay leaf and either push all through a small sieve or you can do like I did, and dump it into the food processor and pulse until all the big chunks of veggies are broken up.
Transfer back into the pot and bring back to a slow boil.
When serving, add a pinch of paprika to the top and enjoy!
Just an added note, this didn't need any ham, it's pretty hearty, so if you are looking to go meatless, this will do nicely. However, we are carnivores and will be adding ham next time.