  Based on Casas-Alvero conjecture \textit{[J. Algebra, 2001]} we formulate the
following conjecture.\\ \textbf{C*-algebraic Casas-Alvero Conjecture : Let
$\mathcal{A}$ be a commutative C*-algebra, $n\in \mathbb{N}$ and let $P(z) =
(z-a_1)(z-a_2)\cdots (z-a_n)$ be a polynomial over $\mathcal{A}$ with $a_1,
a_2, \dots, a_n \in \mathcal{A}$. If $P$ shares a common zero with each of its
(first) $n-1$ derivatives, then it is $n^\text{th}$ power of a linear monic
C*-algebraic polynomial.}\\ We show that C*-algebraic Casas-Alvero Conjecture
holds for C*-algebraic polynomials of degree 2.
