The chromatic number $\chi(X)$ of a topological space $X$ is related to the separation dimension $t(X)$ introduced and studied by Steinke.
The separation dimension $t(X)$ is defined inductively:
$\bullet$ $t(\emptyset)=-1$
$\bullet$ $t(X)=0$ for any space $X$ of cardinality $|X|=1$;
$\bullet$ if $|X|\ge 2$, then $t(X)\le n$ if for each subspace $M\subset X$ with $|M|\ge 2$ there exists a set $A\subset M$ such that $t(A)<n$ and $X\setminus A$ is disconected.
It is easy to see that $t(X)=0$ if and only if the space $X$ is totally disconnected.
In Proposition 3.1 of his paper Steinke proved the following
Sum Theorem: For any subspaces $A,B$ of a topological space the union $A\cup B$ has separation dimension $t(A\cup B)\le t(A)+t(B)+1$.
This theorem implies that $t(A)+1\le\chi(X)$.
On the other hand, by the classical Decomposition Theorem of Urysohn (this is Theorem 7.3.9 in Engelking's book "General Topology"), for a metrizable space $X$ of finite dimension $Ind(X)$ the number $Ind(X)+1$ is equal to the smallest partition of $X$ into subsets of large inductive dimension zero.
Since spaces of large inductive dimension zero are totally disconnected, this decomposition theorem implies that $\chi(X)\le Ind(X)+1$ for any metrizable space $X$. Therefore, for any metrizable space $X$ of finite large inductive dimension, we obtain the inequalities:
$$t(X)+1\le \chi(X)\le Ind(X)+1.$$
In Corollary on page 279 of his paper, Stainke proves that for each locally compact paracompact space $X$ we have the inequalities
$$dim(X)\le t(X)\le ind(X)\le Ind(X).$$
Since $dim(X)=ind(X)=Ind(X)$ for any separable metrizable space $X$, we finally conclude that
$$dim(X)=t(X)=ind(X)=Ind(X)\quad\mbox{and}\quad\chi(X)=\dim(X)+1$$
for any locally compact separable metrizable space $X$.
In particular,
$$\chi(\mathbb R^n)=n+1,$$
which answers the question of N. de Rancourt.