Place dried split peas in a large soup pot, covering them completely with water. Cover the pot and allow to soak overnight (or for at least 8 hours).
Drain the peas, rinse and return to the pot.
Add 4 cups water, chicken broth, ham bone, onion, bay leaf, salt and pepper to the soup pot. Bring to a boil, and then reduce to a simmer, cover and cook for 90 minutes, stirring occasionally. Remove bay leaf from pot and discard.
Remove the ham bone from the pot and cut off the ham, dicing it into a bite size pieces. Return the ham back into the pot.
Add carrots, potatoes and celery into the pot, bring to a low simmer and continue to cook for 35-40 minutes, or until the vegetables are softened.
Serve soup in bowls, alongside some crusty bread or rolls for dipping.