The Fifth Gamma Matrix This is regarding $\gamma^5$, the fifth gamma matrix in quantum field theory. I know its defining properties, namely,
$$\gamma^5= -i\gamma^0 \gamma^1 \gamma^2 \gamma^3 $$
with $\{\gamma^5,\gamma^{\mu}\}=0$ and $(\gamma^5)^2=1$, but so far all I've used these for is to help prove some gamma matrix identities. 
It seems obvious to me that $\gamma^5$ is dependent on our representation (e.g. Weyl or Dirac), so my original idea that it's useful because of invariance on choice of basis seems wrong. 
What do we like about $\gamma^5$?
 A: One reason why $\gamma^5$ is important is because it corresponds to a symmetry of the massless Dirac Lagrangian:
$$
\mathcal{L} = i \bar{\psi} \gamma^\mu \partial_\mu \psi
$$
and the global transformation:
$$
\psi \to \psi' = e^{i \alpha \gamma^5} \psi \tag{1}
$$
Then using the identity:
$$
\{ \gamma^5,\gamma^\mu \} = 0
$$
we can show:
$$
\gamma^\mu e^{i \alpha \gamma^5} = e^{-i \alpha \gamma^5} \gamma^\mu
$$
Therefore, it is easy to see that:
$$
\bar{\psi} \to \bar{\psi}' = \bar{\psi} e^{ i \alpha \gamma^5}
$$
Thus, we see that the transformation expressed by equation $(1)$ is a symmetry of the massless Dirac Lagrangian:
$$
\mathcal{L} \to \mathcal{L}' = \mathcal{L}
$$
which is known as the axial symmetry. This symmetry will play an important role in the Standard Model and its anomalies. I'm sure you will study this at some point in the future.
Edit:
As innisfree mentions in the comments, using the $\gamma^5$ matrices, we can form the Lorentz invariant projection operators:
$$
P_{\pm} = \frac{1}{2} ( \mathbb{1} \pm \gamma^5)
$$
such that:
$$
P_+ P_+ = P_+
$$
$$
P_- P_- = P_-
$$
$$
P_+ P_- = P_- P_+ = 0
$$
(Note that the above three equations are valid regardless of the representation used.) In the chiral representation, they act on the Weyl spinor as:
$$
P_+ \psi = \begin{pmatrix} 0 \\ \psi_R \end{pmatrix}
$$
$$
P_- \psi = \begin{pmatrix} \psi_L \\ 0 \end{pmatrix} 
$$
In addition, for an any representation of the Clifford algebra, we use $\gamma^5$ to define the left- and right-handed parts of the field $\psi$:
$$
\psi_R  = P_+ \psi
$$
$$
\psi_L = P_- \psi
$$
A: In the standard model, and more generally in quantum field theory, the fifth gamma matrix has several uses. The charge current interaction between $\psi_{u,d}$, $\psi_{d,u}$ and a $W^{\pm}_\mu$ carries a factor of
$$V = \frac{ig}{\sqrt{2}}\gamma_\mu \frac{1-\gamma^5}{2}$$
In addition, the fifth gamma matrix can be used to construct Lagrangians with pseudo-scalars, e.g. $\bar{\psi}\gamma^5 \psi$, or axial vectors such as $\bar{\psi}\gamma^5 \gamma^\mu \psi$.
