There's no such thing in CA as common law marriage. In CA you can have a "Marvin" claim (Marvin v. Marvin, 18 Cal. 3d 660 (1976), which is a type of contract claim between two adults for something other than sex.
In your case, if you move in with someone, you may be entitled to less support because you'll need less by co-habitating. That's not any recognition of that relationship so much as it is a practical assessment that you don't need as much per month to live on if someone else besides you is making the mortgage or rent payments.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.
Attorney Koslyn is correct. Technically there is no common law marriage in California.
Your ex will have to take you to court to modify it based upon your total economic situation. So, if you are moving in with your partner, but your finances are remaining the same by the way you have things set up (i.e. all finances separate, etc) it will be hard for him/her to modify the support.
This is general information only. Good luck with your situation.