Let us assume that we have $f:\mathbb{R} \to \mathbb{R}$. Also let us assume that $x_1\in \mathbb{R}$ and $x_2\in \mathbb{R}$ are given too. With this we can calculate $y_1 = f(x_1)$ and $y_2 = f(x_2)$.
Now if we want to interpolate/extrapolate an in-between value we could take a value like e.g. $\alpha \in \mathbb{R}$ (but for interpolation $\alpha$ would be in range $[0,1]$).
With this we could calculate the interpolated values $x_{irp},y_{irp} \in \mathbb{R}$ like:
$$x_{irp}=\alpha \cdot x_1 + (1- \alpha) \cdot x_2$$ $$y_{irp}=\alpha \cdot y_1 + (1- \alpha) \cdot y_2$$
Now my question:
Does the same approach also works for complex numbers too? In this case instead of $\mathbb{R}$ we would have $\mathbb{C}$.
(I know indeed that for multidimensional input and output this approach works, but for complex numbers too?)
Yes, exactly the same method will work (with exactly the same limitations that this is a really bad estimate in general).