Let us consider $\mathbb{C}^n$, the set of all $n$-tuples of complex numbers. We know that $(\mathbb{C}^n,+,\cdot)$ is a unital commutative complex algebra.
Suppose that $\diamond$ is an associative multiplication on ($\mathbb{C}^n,+$) such that $(\mathbb{C}^n,+,\diamond)$ forms a unital commutative complex algebra.
Q. Suppose that $A:(\mathbb{C}^n,+,\diamond) \to (\mathbb{C}^n,+,\cdot)$ is a unital homomorphism. Is $A$ necessary invertible?
No. The map $\mathbb C^2\to\mathbb C^2$ (with componentwise operations both times), $(z,w)\mapsto (z,z)$ is a counterexample.
Of course, this is just the projection map $\mathbb C^2\to\mathbb C^1$ in mild disguise.