If you move to another state can you keep your health insurance?

You definitely want to check with your insurance carrier first to ask about whether you can keep your plan, and IF you keep your plan whether you will have to go through the underwriting process again.

But also, it depends to a huge degree on the state where you're moving. Every state has its own variation of consumer protection and its own commitment (or lack thereof) to making sure all residents have insurance.

Some states have guaranteed issue (and varying versions of it) which means they're required to sell you a policy. (It doesn't mean the policy will be inexpensive, however.)

States now also have the pre-existing condition insurance plan (PCIP) - the federal risk pool program - and 34 states have their own risk pool plans. These, again, all vary in price and benefits, waiting periods, etc. You need to do your homework, which may include contacting the new state's consumer assistance program.

It's all in the attached link.