First let me give some relevant information:
For every $n$ every subgroup of $GL_n(\mathbb C)$ is isomorphic to a subgroup of $GL_m(\mathbb R)$ for some $m$.
Let $\rho_n: M_n(\mathbb C) \hookrightarrow M_{2n}(\mathbb R)$ be the map $$ \rho_1 (a + ib) = (\begin{array}{cc} a & b \\ -b & a\end{array})$$ and similarly for $n>1$.
A real matrix $A \in M_{2n}(\mathbb R)$ is called complex linear if it is in the image of $\rho_n$. Then there is the following theorem:
If $f_n:\mathbb C^n \to \mathbb R^{2n}$ denotes the map $(x_1 + y_1i, x_2 + y_2i, \dots) \mapsto (x_1,y_1,x_2,y_2, \dots)$ then a real matrix $A \in M_{2n}(\mathbb R)$ is complex linear iff the map $F=f_n^{-1}Af_n$ is linear.
The text I'm reading goes on to define
$$ J_{2n}=\rho_n(i I)$$
and then states "...The matrix $J_{2n}$ is called the standard complex structure on $\mathbb R^{2n}$"
My goal is to, eventually, prove the theorem I posted above but before I can attack that I really need to see what's going on here. Even looking at $J_4$ I don't know what to make of this. To help me understand my question therefore is:
If $J_{2n}$ is the standard complex structure, what would be a non-standard complex structure?
Let $V$ be an even-dimensional real vector space. An (almost) complex structure on $V$ is an endomorphism $\mathcal{J} : V \to V$ such that $\mathcal{J}\circ\mathcal{J} = -\operatorname{id}_V$.
If $V = \mathbb{R}^{2n}$, then with respect to the standard basis we have $\mathcal{J}(v) = Jv$ for some $J \in GL(2n, \mathbb{R})$. As $\mathcal{J}\circ\mathcal{J} = -\operatorname{id}_V$, $J^2 = -I$ where $I$ is the $2n\times 2n$ identity matrix. The form of the standard (almost) complex structure on $\mathbb{R}^{2n}$ depends on how $\rho_n$ is defined ("and similarly for $n > 1$" does not make it clear what $\rho_n$ is). Usually one takes
$$J_{2n} = \left[\begin{matrix} 0 & -I\\ I & 0\end{matrix}\right]$$
where $I$ is the $n\times n$ identity matrix (although it seems, according to your conventions, you may want to swap $-I$ and $I$). Then a non-standard (almost) complex structure on $\mathbb{R}^{2n}$ is given by a matrix $J \neq J_{2n}$ with $J^2 = -I$. For example, any matrix similar to $J_{2n}$ defines a non-standard (almost) complex structure.
Explicit Example: Let
$$J = \left[\begin{matrix} -3 & -5\\ 2 & 3\end{matrix}\right],$$
then $J^2 = -I$, so $\mathcal{J}(v) = Jv$ is a non-standard (almost) complex structure on $\mathbb{R}^2$.
Note, what you've called a complex structure, I've called an almost complex structure. The reason I use the latter is that one can extend the notions above to even-dimensional manifolds where there is already a notion of complex structure. You may also see the phrase linear complex structure; for example, this is the terminology used on Wikipedia.