  We introduce a new class of holomorphic polynomials extending the classical
Gould--Hopper to two complex variables. The considered polynomials include the
$1$-D and $2$-D holomorphic and polyanalytic It\^o--Hermite polynomials as
particular cases. We emphasize studying their operational representation,
various generating functions, and recurrence relations. We also establish some
special identities including multiplication and addition formulas of Runge
type, as well as the Nielson type formulas. Higher-order partial differential
equations are analyzed and the connection to Gould-Hopper polynomials and
hypergeometric functions are investigated.
