If we let $\gamma$ be the circline path from $ 0$ to $1$, how do we list all possible values of
$$\int_{\gamma} z^3dz$$
One of which, I think, could be over the real axis, s.t. $$\int_{\gamma} z^3dz = \int_{0}^1 t^3dt = \frac{1}{4}$$ Another could be the closed path starting at $0$, but how do I initiate this integral?
Is the top integral correct?
First of all, for a fix path $\gamma$ from any point $z_0$ to $z_1$, $$\int_{\gamma}z^3dz$$ is well defined.
But more is true, since your function $f(z)=z^3$ is entire (holomorphic everywhere), the path integral does not depend on the choice of path.
If you have a function with some singularities, you can still define the integral over some path (that doesn't pass through any singularities). Then, for any two homotopic paths $\gamma_0$ and $\gamma_1$ from $z_0$ to $z_1$, you have
$$\int_{\gamma_0}f(z)dz=\int_{\gamma_1}f(z)dz.$$