Geometric interpretation of complex inner products Thanks for reading.
In short, my question is this: 
What's the geometric interpretation of inner products when we allow our vectors to have complex components?

Now, to give a little context...
When I think of vectors that allow for complex components, I think of it as we're attaching a new orthogonal axis to each axis of the cartesian plane in which we put that corresponding complex component.
So $2$ dimensional vectors that allow for complex components are really $4$ dimensional, since both their $x$ component and their $y$ component have an additional "complex axis" attached to them.
I'm not sure if this is a good way to think about vectors with complex components, but it's the way I've been thinking about them thus far.
The dot product between two real-valued vectors $\vec{a}\cdot\vec{b}$ can be interpreted as multiplying the magnitude of the projection of $\vec{a}$ onto $\vec{b}$ by the magnitude of $\vec{b}$.
If we allow $\vec{a}$ and $\vec{b}$ to be complex, then the above only seems to be true when the components of $\vec{a}$ and the components of $\vec{b}$ point in the same "direction" on their corresponding complex planes. 
An example:
Let $\vec{a}=\begin{bmatrix}
i\\ 
1
\end{bmatrix}$ and $\vec{b}=\begin{bmatrix}
i\\ 
-3
\end{bmatrix}$
Then, $\vec{a}\cdot\vec{b}=(i)(-i)+(1)(-3)=(1)-3=-2$
Which can indeed be interpreted as the magnitude of the projection of $\vec{a}$ onto $\vec{b}$ multiplied by the magnitude of $\vec{b}$.
Another example:
Let $\vec{a}=\begin{bmatrix}
2+i\\ 
1
\end{bmatrix}$ and $\vec{b}=\begin{bmatrix}
4+2i\\ 
-3
\end{bmatrix}$
Then, $\vec{a}\cdot\vec{b}=(2+i)(4-2i)+(1)(-3)=(8+2)-3=7$
Once again, this can be interpreted as the magnitude of the projection of $\vec{a}$ onto $\vec{b}$ multiplied by the magnitude of $\vec{b}$.
Note that $(2+i)$ and $(4+2i)$, the two $x$ components of our vectors, point in the same direction on the $x$ complex plane.

However, now let:
$\vec{a}=\begin{bmatrix}
i\\ 
1
\end{bmatrix}$
...and...
$\vec{b}=\begin{bmatrix}
2+3i\\ 
1+i
\end{bmatrix}$
...then $\vec{a}\cdot\vec{b}=4+i$.
We got a complex number as a result...how am I supposed to geometrically interpret that?
Thanks!!!

Think of vectors $z,w \in\Bbb C^n$ (in your case, $n=2$). If you write $z=a+bi$ and $w=c+di$, with $a,b,c,d\in\Bbb R^n$, then you can identify $z$ with the vector $\tilde z=(a,b)\in\Bbb R^{2n}$, and so on. When you compute the hermitian inner product $\langle z,w\rangle$ (which you wrote with dot), you end up with
$$\langle z,w\rangle = (a\cdot c + b\cdot d) + i(b\cdot c - a\cdot d).$$
The real part is the real dot product $\tilde z\cdot\tilde w$. The imaginary part is skew-symmetric (meaning that you get the negative when you switch the order of $z$ and $w$); indeed, it can be written naturally as an alternating $2$-tensor, called the Kähler form of $\Bbb C^n$. It is naturally computing (perhaps up to a constant factor) the signed area of the parallelogram in $\Bbb R^{2n}$ spanned by $\tilde z$ and $\tilde w$.
