The correct definition of higher-dimensional quasiconformal maps does not use complex variables. The correct condition is $$ |Df(x)|\le K |J_f(x)| $$ where $J_f$ is the Jacobian determinant. An orientation-preserving homeomorphism $f$ between two domains $U, V$ in $R^n$ is called quasiconformal if it belongs to the Sobolev class $W^{n,1}_{loc}(U)$ (hence, has 1st order partial derivatives a.e. in $U$) and there exists a constant $K$ such that the above inequality is satisfied a.e. in $U$. There are many other alternative definitions. For instance, instead of working with Sobolev spaces, you can assume that $f$ is absolutely continuous on almost every coordinate line segment in $U$ (hence, has partial derivatives a.e. in $U$) and satisfies the same inequality as above. if you do not know about Sobolev spaces or absolute continuity, just think of $f$ as a diffeomorphism (this is not enough, but suffices for the intuition).
Other definitions are in terms of conformal moduli, conformal capacities, quasisymmetry,...
Some references:
Iwaniec, Tadeusz; Martin, Gaven, Geometric function theory and nonlinear analysis, Oxford Mathematical Monographs. Oxford: Oxford University Press (ISBN 0-19-850929-4/hbk). xv, 552 p. (2001). ZBL1045.30011.
Reshetnyak, Yu. G., Space mappings with bounded distortion, Translations of Mathematical Monographs, 73. Providence, RI: American Mathematical Society (AMS). xv, 362 p. (1989). ZBL0667.30018.
Väisälä, Jussi, Lectures on (n)-dimensional quasiconformal mappings, Lecture Notes in Mathematics 229. Berlin-Heidelberg-New York: Springer-Verlag. XIV, 144 p. (1971). ZBL0221.30031.
Addendum. The analogue of the Beltrami differential of a map $f$ in higher dimensions is $$ M_f(x):= J^{-2n}_f(x) (Df(x))^T Df(x), $$ a field of symmetric positive-definite matrices on $U$. The Beltrami equation $$ M_f(x)=A(x), $$ with $A(x)$ a field of positive-definite symmetric matrices on $U$, is overdetermined if $n\ge 3$ (as Alex noted in his answer). Nevertheless, this equation is sometimes useful when working with quasiconformal maps, although not as useful as the classical Beltrami equation: Most tools in higher dimensions are not analytic but geometric. An interesting thing is that if $A(x)$ is smooth, then there are known necessary and sufficient conditions for solvability of the higher-dimensional Beltrami equation (the condition is a 3rd order nonlinear PDE on $A$ if $n=3$ and a 2nd order nonlinear PDE on $A$ if $n\ge 4$). I do not know if anybody worked out a distributional analogue of this classical result.