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.