$\mathcal A$ is a Banach algebra. Can there be two different $*$ operators which both make $\mathcal A$ a $C^*$-algebra? I really have no idea where to start. The only thing I know is that there can not be different $C^*$-norms (whether complete or not) on a $C^*$-algebra, but I find that I barely know nothing about algebraic isomorphisms between $C^*$-algebras.
If such isomorphism (which is not a $*$-isomorphism) exists, it must map some self-adjoint element to a non self-adjoint one. I can neither find an example nor show it is impossible.

The answer pointed out by @Mark presumes that you want to keep the norm, that is, there is no adjoint operation
"$^{\bigstar}$" on $\mathcal A$, other than the defaul adjoint, such that
$(\mathcal A, ^{\bigstar}, \|\cdot\|)$
is a
C*-algebra, where $\|\cdot\|$ is the default norm.
However,  it is possible to find a different adjoint operation "$^{\bigstar}$", and a different norm $|||\cdot|||$,
such that
$(\mathcal A, ^{\bigstar}, |||\cdot|||)$
is a C*-algebra.
All you need to do is choose an automorphism
$$
  \varphi :\mathcal A \to \mathcal A,
  $$
which preserves everything but the star and norm,  and define a new star and norm by
$$
  a^\bigstar := \phi^{-1}\big (\phi(a)^*\big ), \quad \text{and} \quad |||a||| := \|\phi(a)\|.
  $$
One such automorphism may be taken to be
$$
  \phi(a) = uau^{-1},
  $$
where $u$ is a non-unitary, invertible element, such that $u^*u$ is non-central.

One more point: on a commutative C*-algebra one cannot find another adjoint operation, even if one is willing to consider a
change of norm.  The reason is that self-adjoint elements may be characterized as those with real spectrum and,
moreover, the space of self-adjoint elements determines the adjoint operation.
