You will put yourself in a much better position for the best California car insurance rates if you have more than one policy with the same provider. This definitely implies you should endeavor to purchase your auto policy from the same insurer that handles your home, medical or other insurance policies. 

Nevertheless, note that you may be better served if you buy your various policies from different providers who offer the cheapest rates for each policy. Do some research with quotes sites and also discuss with your agent. This done, you will be be well equipped to determine if staying put would make you save a lot more.