  We consider imaginary Verma modules for quantum affine algebra
$U_q(\hat{\mathfrak{sl}(2)})$ and construct Kashiwara type operators and the
Kashiwara algebra $\mathcal K_q$. We show that a certain quotient $\mathcal
N_q^-$ of $U_q(\hat{\mathfrak{sl}(2)})$ is a simple $\mathcal K_q$-module.
