This was a question on my exam.

Is the function

f : C → C,

z = a + ib → f(z) = a + i(a + b)

of the form f(z) = z + w where w does not depend on z?

The answer to this question was YES, because w=ai. But a is det real part of the complex number z, so why is it correct to say that w does not depend on z? Any one who can explain it to me?