Is there any medicine can cure eczema and is there any cream or ointment that can remove the dark pigments caused by eczema?

No cure for eczema: But topical steroids can help a lot. For the hyperpigmentation, there are bleaching creams, but they can also irritate, especially if you have eczema. The best way is to control the underlying eczema, and the dark spots will eventually fade. See a dermatologist to make sure that your skin condition is nothing more serious.