Higher dimensional analogues of the argument principle?

Of course, the Argument Principle is in the context of meromorphic functions. However, the higher-dimensional analogue in the context of smooth functions comes from the theory of the degree of maps and winding numbers.

In particular, suppose $W$ is a compact $n$-dimensional oriented manifold with boundary, $f\colon W \to \Bbb R^n$ is smooth, and $f\big|_{\partial W} \ne 0$. If $0$ is a regular value of $f$, then $f^{-1}(0)$ consists of a finite number of points $x_1,\dots,x_k\in W$, each of which appears with a sign $\epsilon_j$ ($+1$ if $df_{x_j}\colon T_{x_j}W\to\Bbb R^n$ is orientation-preserving, and $-1$ if it is orientation-reversing). Then $$\sum_{j=1}^k \epsilon_j = \text{deg}\left(\frac{f}{\|f\|}\Bigg|_{\partial W}\colon \partial W\to S^{n-1}\right)\,.$$

Comment: In the case $n=2$, $W\subset\Bbb R^2$, and $f$ holomorphic, the $\epsilon_j$ are all always $+1$, and so we merely count the roots in $W$. The degree on the right-hand side is the winding number of $f(\partial W)$, which is exactly what $\dfrac1{2\pi i}\displaystyle\int_{\partial W} \frac{f'(z)}{f(z)}dz$ computes.

The best reference I know on such matters is Guillemin and Pollack's Differential Topology. See pp. 110-111 and 144, in particular.

EDIT: To respond to your additional comments/questions, the one-variable calculus definition of derivative extends to $\Bbb C$ only because $\Bbb C$ is a field. Once you move on to $\Bbb R^n$, the derivative becomes a linear transformation with a limit property but cannot itself be given by the sort of limit you desire. Only directional derivatives have the single-variable calculus definition. (In time, you will learn some rigorously done multivariable calculus.) The degree calculation I have given above can indeed be represented by an integral (but, once again, you'll have a bit to learn to understand this): Choose an $(n-1)$-form $\omega$ on $S^{n-1}$ with $\displaystyle\int_{S^{n-1}}\omega=1$, and compute $\displaystyle\int_{\partial W} \left(\tfrac{f}{\|f\|}\right)^*\omega$. Alternatively, compute $$\tfrac1{\text{vol}(S^{n-1})}\int_{\partial W} f^*\left(\tfrac{x_1\, dx_2\wedge\dots\wedge dx_n - x_2\, dx_1\wedge dx_3\wedge \dots \wedge dx_n + (-1)^{n-1} x_n\, dx_1\wedge\dots\wedge dx_{n-1}}{(x_1^2+x_2^2+\dots+x_n^2)^{n/2}}\right)\,.$$ This is in fact an immediate generalization of the integral appearing in the argument principle.

There are generalizations of the Residue Theorem (which is where the Argument Principle comes from) to meromorphic functions in $\Bbb C^n$. This is also a rather rich subject; some comments were made in this post.