Is the residue of a complex function always real?

Let $f(z) = \frac{H(z)}{(z-z_o)^m}$, where H(z) is an analytic function, the residue of f(z) is $Res(f,z_o) = c_{m-1}$

But recall that the residue formulas (integral form) has a $2\pi i$ factor, so it is quite surprising that in most examples I find that the residue is a real number

So I wonder if the residue can be referred to as the "real part" of the principal part of the Laurent series of a complex function.

Can someone help me validate whether the above claim is true?

The residue of a Laurent series $f(z) = \sum_{n \in \mathbb{Z}} c_n z^n$ is the coefficient on its $z^{-1}$ term. If we have a function with real residue, i.e. $c_{-1} \in \mathbb{R}$, how does the residue of e.g. the function $i f(z)$ compare?