There is the continuum 2-dimensional Gaussian free field, which is a higher dimensional generalization of Brownian motion. The continuum GFF satisfies the so-called domain Markov property (meaning if you condition on the value of the GFF on a subset of $\mathbb{R}^2$, its value outside that subset depends only on the value on the boundary of that subset), and can be viewed as weak limit of GFF defined on 2-dimensional infinite lattices. The discrete GFF is basically a probability distribution on $\mathbb{R}^{\mathbb{Z}^2}$, i.e., the space of real-valued functions on $\mathbb{Z}^2$. The distribution is given in terms of a Hamiltonian,
$$\displaystyle H(x) = \sum_{(i,j) \in E} x_i x_j + \sum_{i \in \mathbb{Z}^2} x_i^2 $$
In other words, the probability density is proportional to $\exp(-H(x))$.
Thus it's the most natural Gaussian measure on $\mathbb{R}^{\mathbb{Z}^2}$ that takes into account the underlying graph structure.

As an analogy, the Brownian motion at discrete time points, say $\mathbb{Z}$ is the Gaussian free field on $\mathbb{Z}$. Another perspective is to view Gaussian free field as the standard Gaussian random variable on the $\mathbb{Z}^2$ or $\mathbb{Z}$ but with the Dirichlet inner product instead of the usual Euclidean inner product; since Dirichlet product is basically the $\mathcal{L}^2$ inner product of the gradient, one needs to impose boundary condition or define certain equivalence classes of functions in order for the inner product to be nondegenerate.

When we take finer and finer grid in $\mathbb{Z}$ or $\mathbb{Z}^2$, we obtain in the weak limit the Brownian motion and the continuum 2-dimensional Gaussian free field. The only thing weird about the continuum GFF in 2 dimensions is that it's no longer a probability law on a function space, but rather on the space of distributions, or generalized functions on $\mathbb{R}^2$; Brownian motion on the other hand can still be viewed as a distribution on continuous functions on $\mathbb{R}$. For more details on how to define such a limiting object, see the lecture notes by Scott Sheffield: Gaussian free fields for mathematicians.