Suppose that I'm dealing with a function of several complex variables, holomorphic in each variable separately.
Should I expect that the contour deformation will work in essentially the same way as in the one variable case? In particular, do I get Cauchy's integral formula for general contours as in the one-dimensional case?
Would be grateful for some references!
The simplest form of multidimensional Cauchy formula is written for polydisks. Let $(r_1,\ldots,r_n)$ be positive radii and consider the compact submanifold $$P=\{(z_1,\ldots, z_n)\in \mathbb{C}^n, |z_1|=r_1,\ldots, |z_n|=r_n\}.$$
Then Cauchy's formula will tell you that the mean value of $f$ on $P$ is $f(0)$. More generally, from the Poisson kernel you can recover $f$ on the inside of $P$ given its values on $P$. See https://en.wikipedia.org/wiki/Cauchy%27s_integral_formula#Several_variables
If I understand correctly, what you also want is a form of contour integral, that is, a generalisation of $\int_C f dz=0$ where you integrate "along a curve".
To do this, suppose first that your contour $Z$ can be sliced: for each $(z_2,\ldots,z_n)$, the intersection $Z\cap (\mathbb{C}\times (z_2,\ldots, z_n))$ is a contour (or a collection of such). Then, for each $z_2,\ldots, z_n$ you can write the usual contour integral, then integrate it over $z_2,\ldots, z_n$. At the end, you will recover something like $$\int_Z f(z)t_1(z)=0,$$ where $t_1(z_1,\ldots, z_n)$ is the unit tangent vector (a complex number !) to $Z\cap (\mathbb{C}\times (z_2,\ldots, z_n))$ at $z_1$.
Depending on your problem, $t_1$ might not be enough to conclude; of course you can apply a unitary change of variables and cut $Z$ along arbitrary slices. But be careful: when cutting $Z$ along complex planes you want to recover contours. If you want it to work for any complex plane, $Z$ needs to be totally real. Under this hypothesis, you have a vector-valued inequality: $$\int_Z f(z)\overrightarrow{t}(z)=0,$$ where $\overrightarrow{t}(z)=(t_1(z),\ldots, t_n(z))$.