Need to add a new field on "Complete Registration" form but I can't:

		/* Build the form */
		$form = new \IPS\Helpers\Form( 'form', 'register_button' );
		if( !\IPS\Member::loggedIn()->real_name OR \IPS\Member::loggedIn()->name === \IPS\Member::loggedIn()->language()->get('guest') )
		{
			$form->add( new \IPS\Helpers\Form\Text( 'username', NULL, TRUE, array( 'accountUsername' => \IPS\Member::loggedIn() ) ) );
		}

		if( !\IPS\Member::loggedIn()->email )
		{
			$form->add( new \IPS\Helpers\Form\Email( 'email_address', NULL, TRUE, array( 'accountEmail' => TRUE ) ) );
		}

		$form->addButton( 'cancel', 'link', \IPS\Http\Url::internal( 'app=core&module=system&controller=register&do=cancel', 'front', 'register' )->csrf() );

		\IPS\Output::i()->output = \IPS\Theme::i()->getTemplate( 'system' )->completeProfile( $form );

You can.

\IPS\Output::i()->output = \IPS\Theme::i()->getTemplate( 'system' )->completeProfile( $form );

You got lucky! The form is being passed in as an object instead of as (string)$form. Do a theme hook on front:system, then in PHP mode, overload the completeProfile method. You should be able to make changes to the form in there, then call the parent.