Morita equivalence of algebras certainly don't preserve commutativity: even if $A$ is commutative there are plenty of noncommutative algebras which are Morita equivalent with $A$---for example all algebras of the form $M\_n(A)$ are good. What is the simplest example of an algebra which is not Morita equivalent to any commutative algebra? 



Purely noncommutative algebra-Morita equivalence
An algebra is Morita equivalent to a commutative algebra iff it's Morita equivalent to its center, since the center is Morita invariant. So any representative of a nontrivial class in the Brauer group of the underlying field $k$ is a counterexample: for example, when $k = \mathbb{R}$ we can take the quaternions $\mathbb{H}$. 


