This question says to regard the set of complex numbers as a vector space over the reals
and to find a linear transformation to which is not complex linear.
What does complex linear mean?
the complex numbers are also a field, so we have 2 choices for regarding them as a vector space:
1) as a vector space over the complex field, of dimension 1, with basis {1}
2) as a vector space over the real field, of dimension 2, with basis {1, i}.
you are being asked to find a mapping which is linear in the sense of (2), but not in the sense of (1).