if we have $\alpha$ a complex function, and we want to take the Hodge dual of
$$d\bigg(\frac{1}{|\alpha|^2}\bigg)d(\bar{\alpha})$$ what will that give us? Can we take $$\star\bigg(\frac{1}{|\alpha|^2}\bigg)d(\bar{\alpha}) + d\bigg(\frac{1}{|\alpha|^2}\bigg)\star d(\bar{\alpha})$$ or this doesn't make sense at all?