given two complex vectors, what is the geometric interpretation of their dot product?
$$ \mathbf{x}\mathbf{y} = \sum x_i y_i^* $$
is there any interpretation similar to the case with real vectors based on projection of one vector onto another or similar?
If ${\bf x}={\bf a}+{\bf b}i$, ${\bf y}={\bf c}+{\bf d}i$ in $C^n$, define vectors ${\bf u}=({\bf a},{\bf b})$, ${\bf v}=({\bf c},{\bf d})$, ${\bf w}=(-{\bf d},{\bf c})$ in $R^{2n}$. Then $$ {\bf x}\cdot {\bf y} = {\bf u}\cdot {\bf v}+i{\bf u}\cdot {\bf w} $$ so the real and imaginary parts of ${\bf x}\cdot {\bf y}$ have exactly the geometric interpretations you are familiar with, as applied to ${\bf u}$, ${\bf v}$ and ${\bf w}$, where ${\bf w}$ is a rotation of ${\bf v}$.
But I think it is best to separate some of the concepts a bit. If we take "geometry" to refer to concepts of distance and angle, then projection goes beyond geometry because it involves linear combinations; we don't really project a vector to a vector, instead we project it to a subspace. And, while it is true that the projection of ${\bf x}$ to the span of ${\bf y}$ in $C^n$ gives the point of the subspace closest to ${\bf x}$, the word to focus on is not so much "closest" (geometry), but "subspace" (algebra).
The simplest example of this is probably in $C^1$ versus $R^2$: The projection of $x=a+bi$ onto the span of $1$ is $x$ itself. But the projection of $(a,b)$ onto the span of $(1,0)$ is $(a,0)$. In both cases you have the closest point; what has changed is the subspace rather than the concept of distance.
