for "operation-preserving" to make sense, you have to say WHICH operation on C, h is preserving. in this particular case, it is clear to me the operation is "complex multiplication".
in particular, since C is a field, the usual laws of arithmetic apply (we can "FOIL"), so:
(a+bi)(c+di) = ac + a(di) + (bi)c + (bi)(di) = ac + (ad)i + (bc)i + (bd)i2.
now in the complex numbers, we have i2 = -1, so:
ac + (ad)i + (bc)i + (bd)i2 = ac - bd + (ad + bc)i.
what you have to show is the following:
h((a+bi)(c+di)) = h(a+bi)⊕h(c+di), that is:
h(ac-bd + (ad+bc)i) = h(a+bi)⊕h(c+di).
you will have to use the definition of h, and the definition of ⊕.