The Maricopa County Attorney's Office filed the motion, which was widely expected, Wednesday.
If the motion is granted, it means that office will be engaged in prosecuting two death penalty cases of serial killers simultaneously.
Cooksey, a 35-year-old Phoenix man, was released from state prison in July after serving a 16-year sentence for his role in a fatal stick-up at a Phoenix strip club.
Four months after he got out, police say, he began an orgy of violence, killing people he mostly knew. Many had been in court on drug charges. One was the brother of his girlfriend. Two were apparently unknown to him: an armed security guard and a woman who police say he raped, killed, and dumped in an alley in a crime of opportunity.
In its 13-page court filing, county prosecutors cite Cooksey's long criminal history, that his slayings were cruel and inflicted pain and anguish on his victims.