You might have to testify against yourself in court, without Fifth Amendment protection, if you're given immunity for whatever you are about to testify to. (This is interesting, because federal courts can use the testimony from state proceedings against you, but the state can't give you federal immunity. Double jeopardy does not apply because the state and federal courts are different. Pretty sure that's not what the framers meant when they were writing the Tenth Amendment...)

Ever since the Hiibel ruling http://en.wikipedia.org/wiki/Hiibel_v._Sixth_Judicial_Distri..., the states have adopted a hodgepodge of differing identity query standards.