  The first (associative) Weyl algebra is formally rigid in the classical
sense. In this paper, we show that it can however be formally deformed in a
nontrivial way when considered as a so-called hom-associative algebra, and that
this deformation preserves properties such as the commuter, while deforming
others, such as the center, power associativity, the set of derivations, and
some commutation relations. We then show that this deformation induces a formal
deformation of the corresponding Lie algebra into what is known as a hom-Lie
algebra, when using the commutator as bracket. We also prove that all
homomorphisms between any two purely hom-associative Weyl algebras are in fact
isomorphisms. In particular, all endomorphisms are automorphisms in this case,
hence proving a hom-associative analogue of the Dixmier conjecture to hold
true.
