Help with understanding definition and how to visualize spherical basis vectors? Can someone please help me understand the definition and how I can visualize these spherical basis vectors in 3-space?
https://en.wikipedia.org/wiki/Spherical_basis#Spherical_basis_in_three_dimensions
$e_+ = -\frac{1}{\sqrt{2}}e_x - \frac{i}{\sqrt{2}} e_y$
$e_0 = e_z$
$e_- = +\frac{1}{\sqrt{2}} e_x - \frac{i}{\sqrt{2}} e_y$  
Where $e_x$, $e_y$, $e_z$ are the standard, Cartesian basis vectors in 3D.  
So an arbitrary vector $r \in \mathbb{R}^3$ can be expressed as:
$r = a_+ e_+ +  a_- e_- + a_0 e_0$.  
What's most confusing to me is the use of imaginary numbers in $e_+$ and $e_-$. How does one go $i$ units in the $e_y$ direction? And even after I assume that $i$ is in the $e_z$ direction, I'm confused by the use of complex coefficients in the x-y plane. Wouldn't that mean that if we allowed complex coefficients that $Span(e_+) = Span(e_-)$, making $e_+$ and $e_-$ linearly dependent?

$a_+$ and $a_-$ are in general not real for real vectors $r$. The allowed values for them to give a real vector is a subspace of $\Bbb C \times \Bbb C$ of two real dimensions. This is inconvenient for ordinary representations of real vectors, but this particular representation is useful in some situations.
