We are interested to explore the limit in using deep learning (DL) to study the electromagnetic response for complex and random metasurfaces, without any specific applications in mind. For simplicity, we focus on a simple pure reflection problem of a broadband electromagnetic (EM) plane wave incident normally on such complex metasurfaces in the frequency regime of 2 to 12 GHz. In doing so, we create a deep learning (DL) based framework called metasurface design deep convolutional neural network (MSDCNN) for both the forward and inverse design of three different classes of complex metasurfaces: (a) Arbitrary connecting polygons, (b) Basic pattern combination, and (c) Fully random binary patterns. The performance of each metasurface is evaluated and cross-benchmarked. Dependent on the type of complex metasurfaces, sample size, and DL algorithms used, MSDCNN is able to provide good agreements and can be a faster design tool for complex metasurfaces as compared to the traditional full-wave electromagnetic simulation methods. However, no single universal deep convolutional neural network (DCNN) model can work well for all metasurface classes based on detailed statistical analysis (such as mean, variance, kurtosis, mean squared error). Our findings report important information on the advantages and limitation of current DL models in designing these ultimately complex metasurfaces.