In content type show same field twice with different widget
The "right" way to add additional elements to a view mode (without using a module like Display Suite or Panelizer to take over view modes) is to use hook_field_extra_fields():  This hook will make your added element visible on the core view mode configuration screen.

If you're new to Drupal it might be easier to go with the preprocess route. But instead of working at the level of template_preprocess_field() you should be working at the level of template_preprocess_node(). The node hook contains the field you want to duplicate. The field hook deals with only one field at a time.