Please get an attorney in your area that can help you navigate the comp system. There is no "state workers comp" as you refer to except for a quasi state but really private. Private insurance is in almost every case. The short answer is no you can not just go get your own medical doctor. There is a medical provider network for the insurance company that you are a part of unless they don't have that or unless there is some exception that you should be out of it. There are plenty of good doctors in the network that an attorney in your area can help you find. Any other medical care you get privately would be out of your own pocket and there would be arguments about admissibility. Please get an attorney. This is only the beginning of the questions that have so many different ways of answering. Check out caaa.org
You always have the right to see your own doctor, but workers' compensation does not have to pay for that in an accepted claim if they have their own medical provider network. See http://www.wcwebzine.com/a-z/medicaltreatment.htm and the linked topic pages. Also note you can get second opionions under workers' compensation : see http://www.wcwebzine.com/a-z/secondopinion.htm. If you want to rely on a private doctor's reports at trial, you run into admissibility issues. See http://www.wcwebzine.com/magazine/issue1-4-13/admissibleevidence4-13.htm
Unless you have predesignated your doctor to treat you for on the job injuries, for the first 30 days after you report an injury, the insurance company has the right to choose what doctor you see. After that, you can pick the doctor who treats you. However, most insurance companies have set up Medical Provider Networks. While there are exceptions, you ordinarily have to choose from the doctors they have in their network. It is entirely possible that your doctor is in that network. You have the right to get the information necessary to access that list of doctors.
The real question is whether your own doctor is the appropriate doctor to treat you for an industrial injury. California's workers comp system is very complex and full of pitfalls. You need to be treating with a doctor who is not only the best available to treat your injury, but also the best available to navigate the workers comp system on your behalf. Chances are that you don't have enough knowledge of the players and the system to make the right choice. A good workers comp attorney can be an invaluable asset when it comes to making the right decisions about things like this. You can find good ones on this site and at www.CAAA.org, the organization for attorneys who represent injured workers in California.
Because the carrier is AIG, the answer practically speaking, is no. You have to pick a doctor in AIG's network (MPN). Mr. Batchelder and Ms. Harris have also mentioned important considerations.
If you pre-selected your own physician prior to the industrial injury you have the right to see your own physician. But if you did not provide your employer with the document then you will have to see the physician within your employer's Managed Provider Network. You may find that your doctor is on the list. You should speak with an attorney to help you through the maze of the California Worker's Compensation System.