I know that a Linear Transformation, T is a function ( mapping ) from a vector space X to another vector space Y, with the linearity property - T(a.x + b.y) = a.T(x) + b.T(y) ; where 'a' and 'b' are scalars.
Suppose X and Y are complex and real vector spaces respectively. Since X is a complex vector space 'a' and 'b' may be imaginary. So, a.T(x) will not be defined, for T(x) is an element of Y, which is a real vector space ( whose field of scalars is the set of real numbers ).
Using the above reasoning, I suppose that such a linear transformation is not possible.
It's more a question of not making sense. A linear transformation is a special kind of function from a vector space over a field into another vector space over the same field.