I have two answers for you, depending on what you have in mind.
You want to add an $x$ to the ring of 2x2 matrices, such that while $x$ commutes with multiples of the identity, it doesn't commute with anything else. You can adjoin a noncommuting indeterminate by using what's called the free product. You take the two rings $M_2(\mathbb{R})$ and $\mathbb{R}[x]$, and then you form the free product over $\mathbb{R}$. Wikipedia has an entry on the free product of groups. The construction for rings is fairly similar.
That construction has one weakness: $x$ will not satisfy any relations. There relations that all 2x2 matrices will satisfy, but $x$ in the free product won't. Rings all of whose elements satisfy identities are known as polynomial identity rings. For example, any four 2x2 matrices satisfy an identity of degree four (described in this blog post). So any three elements of $M_2(\mathbb{R})$ and $x$ should satisfy that relation. (I don't know if all possible relations are generated from specializing this one relation. There could be other relations that rely on specific properties of specific matrix elements.)
As in Birdman's comment, the motivic fundamental group is unifying the notion of monodromy action on the fibers of local systems of "geometric origin."
To explain this, let us start with the case of a field $K$. We have a semisimple $\mathbb{Q}$-linear Tannakian category $\operatorname{Mot}_K$ of (pure) motives over $K$ for which fiber functors are cohomology theories, i.e., it makes sense to have an $L$-valued fiber functor for a field $L$, and this is the same as a Weil cohomology theory for smooth proper $K$-varieties with values in $L$-vector spaces. A motivic Galois group, to my understanding, is attached to a cohomology theory/fiber functor $F$ of $\operatorname{Mot}_K$.
Then the motivic Galois group is the associated group scheme/$L$ whose representations are given by the category $\operatorname{Mot}_{K}\underset{\mathbb{Q}}{\otimes}L$, i.e., it is the group scheme of automorphisms of the fiber functor $F$. So it is "the group which acts on $F$-cohomology of (smooth projective) varieties." Since this category is semi-simple, the motivic Galois group is pro-reductive. E.g., the absolute Galois group (considered as a discrete group scheme) of $K$ acts on $\ell$-adic cohomology, so there is a homomorphism from $\operatorname{Gal}(K)$ to the motivic Galois group of $K$ corresponding to the fiber functor defined by $\ell$-adic cohomology.
For, say, a smooth variety $X$ over $K$, there should be a category of "motivic sheaves" on $X$, or rather, a semi-simple category of pure motivic sheaves contained in an Artinian category of mixed motivic sheaves. You should have e.g. an $\ell$-adic" fiber functor from the mixed category to $\ell$-adic perverse sheaves on $X$ which sends pure guys to (cohomologically shifted) lisse sheaves (alias local systems). E.g., if $K=\mathbb{F}_q$, then this is the category of pure (resp. mixed) perverse sheaves on $X$. If $K=\mathbb{C}$, this should be a full subcategory of pure (resp. mixed) polarizable Hodge modules on $X$. For any smooth proper (resp. just any) map $f:Y\to X$, there should an object in the category of pure (resp. mixed) motivic sheaves on $X$ corresponding to push-forward of the structure sheaf on $Y$.
The motivic fundamental group act on the ``fibers" of pure motivic sheaves on $X$. I.e., for a $K$-point of $X$, you should get a functor to the category of $K$-motives. This is a motivic incarnation of taking the fiber of a local system. Then given our cohomology theory $F$, we obtain a functor from pure motivic sheaves on $X$ to $L$-vector spaces, and the automorphisms of this functor will be the $F$-realization of the motivic Galois group of $X$.
Best Answer
Assume $X$ is a projective threefold with $n$ ordinary double points. Then $X$ has $2^n$ small resolutions of singularities. Usually, almost of all of them are non-projective.