As long as no one else claims the child as a dependent on another federal return, you could claim that child on your state return. To do that, you would have to either prepare a manual state return or use a TurboTax desktop (CD/Download) program and prepare two returns, one with the child as a dependent on both federal-state return and the other without the child.

The only reason to claim a child on your state return and not on your federal return is if someone else is claiming them for federal purposes.  If the child qualifies as someone else's dependent on their federal return, then only they would be able to claim a dependent exemption on a state return. 

Note that you will NOT be able to file the state return electronically through Turbo Tax in this situation.  I found this out the hard way after I paid the state filing fee.  I went into Turbo Tax and created a second return.  I paid the fee to file the state return electronically.  I only wanted to file the state, but because the federal amount had changed and I had already filed federal, it sent me into a perpetual loop that I could not get out of.  I later found out that Turbo Tax will not let you electronically file a state return without electronically filing a federal return.  I filed the state return electronically with the wrong amount and will now need  to file an amended return with the state.  

Does this hold even in a married filing separately situation?  Both me and my wife can claim our daughter on the federal return, but she claims her as the higher earner. Could I claim her on the state return? 

Yes, usually the child must be on your federal return to claim the state exemption.  States do have some exceptions and you would have to check the state instructions for your state.  Here is an AZ example.  

 

 

The following individuals may qualify as dependents or "Other Exemptions" on your Arizona return but not on your federal return:

1. An individual you could have claimed on your federal return but didn't in order to allow the individual to claim his or her own federal education credit. The individual must qualify as a federal dependent even though you didn't include them on your federal return. For more information on this situation, see Arizona Individual Tax Ruling 05-2 . These individuals will be listed with the rest of your dependents on your Arizona return.
2. An individual 65 years of age or older (as of January 1, 2020) who doesn't qualify on your federal return as a dependent. You must have paid more than one-fourth the cost for a nursing or assisted care facility (costs must exceed $800), or you paid more than $800 for either home health care or other medical costs for the person. These individuals will be listed as "Other Exemptions" on your Arizona return.
3. A stillborn child qualifies if the stillbirth occurred in 2019, you received a certificate of birth resulting in stillbirth from the Arizona Department of Health Services, and the child would have otherwise been a member of your household. These individuals will be listed as "Other Exemptions" on your Arizona return.