var autoComplete,
     dlg = $("#dialog"),
     input = $("#input");

 // initialize autocomplete
 input.autocomplete({
     ...
 });

 // get reference to autocomplete element
 autoComplete = input.autocomplete("widget");

 // init the dialog containing the input field
 dlg.dialog({
      ...
 });

 // move the autocomplete element after the dialog in the DOM
 autoComplete.insertAfter(dlg.parent());

Added: Use .ui-front instead of .zIndex() for the suggestions menu. (7d5fe02)

ul.ui-autocomplete.ui-menu {
  z-index: 1000;
}