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We consider imaginary Verma modules for quantum affinealgebraU_q(\widehat{\mathfrak{sl}(2)}) and define a crystal-like base which wecall an imaginary crystal basis using the Kashiwara algebra K_q constructed inearlier work of the authors. In particular, we prove the existence of imaginarylike bases for a suitable category of reduced imaginary Verma modules forU_q(\widehat{\mathfrak{sl}(2)}).

Imaginary Verma Modules for $U_q(\widehat{\mathfrak{sl}(2)})$ and Crystal-like bases