We study the information entropy, order, disorder, and complexity for the two-dimensional (2D) rotating and nonrotating Bose-Einstein condensates. The choice of our system is a complete theoretical laboratory where the complexity is controlled by the two-body contact interaction strength and the rotation frequency ($Omega$) of the harmonic trap. The 2D nonrotating condensate shows the complexity of the category I where the disorder-order transition is triggered by the interaction strength. In the rotating condensates, $Omega$ is chosen as the disorder parameter when the interaction strength is fixed. With respect to $Omega$, the complexity shifts between the maximum and minimum confirm the existence of category II complexity in the rotating condensate. Also, We consider the interaction strength as the disorder parameter when $Omega$ is unchanged and complexity as a function of interaction strength exhibits category III complexity. The present work also includes the calculation of upper bound and lower bound of entropy for 2D quantum systems.
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