When trying to find synonyms for the word "swivel-eyed," one could consider adjectives such as "wild-eyed," "crazed," or "manic." These descriptors all imply a similar state of emotion or demeanor that suggests a lack of control or stability. Other potential synonyms might include phrases or words that evoke a sense of disorientation or unsteadiness such as "unhinged," "frenzied," or "deranged." While some might use more colorful language such as "bug-eyed" or "off their rocker," it is important to try to use language that is both clear and respectful when discussing others' behavior.