Let's say I have a contour integral on a non-closed contour with starting point $z_0$ and ending point $z_1$. Am I allowed to do a substitution like this? And under what assumptions?
$$\displaystyle \int_{z_0}^{z_1} f (z) \, \mathrm d z = \int_{u^{-1}(z_0)}^{u^{-1}(z_1)} f (u (z))u'(z) \, \mathrm d u $$
For example,
$$\int_\epsilon^T t^{1/2 + i}e^{(-4-i)t}\, \mathrm dt \stackrel{?}{\stackrel{u \leftrightarrow (4+i)t}{\longleftrightarrow} }\int_{4\epsilon+i\epsilon}^{4T+iT}\left({\frac{u}{4+i}}\right)^{1/2 +i}e^{-u}\left({\frac{1}{4+i}}\right)\, \mathrm du$$
where $0 < \epsilon < T \in \mathbb R$.
And the integrand is a contour, it's a smooth map from a real interval to the plane, so the integral is a contour integral with the parameterization $t: t \in [\epsilon..T]$ by definition. At least, I'm pretty sure it matches the definition of a contour integral with a parameterization. I even graphed it:
My thoughts:
Since I'd want want the substitution to move the contour somewhere else without changing the value of the integral, then $u$ would have to be a homotopy; thus a necessary condition would be that the old and new contour should each be in a simply connected domain. I'd guess $u$ might have to be analytic and bijective as well.
Suppose $f$ is analytic in the simply connected region $V$ and $\gamma $ is a contour in $V$ from $z_0$ to $z_1.$ Let $U$ be a region (not necessarily simply connected), and assume $u:U \to V$ is an analytic function whose range contains $\{z_0,z_1\}.$ Choose $w_0,w_1\in U$ such that $u(w_0) =z_0, u(w_1) =z_1.$ Let $\delta$ be a contour in $U$ from $w_0$ to $w_1.$ Then
$$\tag 1 \int_\gamma f(z)\,dz = \int_\delta f(u(w))u'(w)\,dw.$$
Proof: Because $V$ is simply connected, $f$ has an antiderivative $F$ in $V.$ Just from the fundamental theorem of calculus (basic one real variable calculus is meant here), the left side of $(1)$ equals $F(z_1)-F(z_0).$ But notice $f(u(w))u'(w)$ also has an antiderivative–in $U$–namely $F(u(w)).$ That follows from the chain rule for analytic functions. Therefore, again by the FTC, the right side of $(1)$ equals $F(u(w_1)) - F(u(w_0)) = F(z_1)-F(z_0)$ and we're done.
If you look at your example, the integrand $z^{1/2+i}e^{(-4-i)z}$ is analytic in the open right half plane, a simply connected region. The map $u(w)= w/(4+i),$ which is entire, takes $w_0 = (4+i)\epsilon, w_1 = (4+i)T$ to $\epsilon,T$ and $(1)$ guarantees your substitution is valid (as would any analytic substitution respecting the end points).
I am not sure what your paragraph "And the integrand is a contour …" is all about. The range of $f\circ \gamma$ isn't the issue here, at least as far as I can see.