A complex number can be represented by a pair of real numbers with some additional rules for multiplication. Therefore, a complex space of dimension $n$ can be seen as a real space of dimension $2n$, with some additional structure of multiplication (which we may ignore when it's not needed: for example, when drawing).
- It's easy to draw a 1-dimensional complex space, because it is also a 2-dimensional real space, a plane.
- It's next to impossible to accurately draw a 2-dimensional complex space, because it is also a 4-dimensional real space, and humans are not good at visualizing or drawing 4 dimensions.
- Since the 3-dimensional complex space is also a 6-dimensional real space, it is much too large for us to draw.
So, people resort to surrogates, such as Reinhardt diagrams (seen here), or drawing slices of $\mathbb C^n$. These don't represent the space, just some aspects of it.
For the second question: before proving something about a mathematical object, one must define it. How do you define the scalar product on $\mathbb C^3$? Two definitions are I know are: $\langle a,b\rangle=\sum \bar a_j b_j$ and $\langle a,b\rangle=\sum a_j \bar b_j$. These are not the same; some people prefer the first and others prefer the second. There's no accounting for taste.