Across the world, millions of users interact with search engines every day to satisfy their information needs. As the Web grows bigger over time, such information needs, manifested through user search queries, also become more complex. However, there has been no systematic study that quantifies the structural complexity of Web search queries. In this research, we make an attempt towards understanding and characterizing the syntactic complexity of search queries using a multi-pronged approach. We use traditional statistical language modeling techniques to quantify and compare the perplexity of queries with natural language (NL). We then use complex network analysis for a comparative analysis of the topological properties of queries issued by real Web users and those generated by statistical models. Finally, we conduct experiments to study whether search engine users are able to identify real queries, when presented along with model-generated ones. The three complementary studies show that the syntactic structure of Web queries is more complex than what n-grams can capture, but simpler than NL. Queries, thus, seem to represent an intermediate stage between syntactic and non-syntactic communication.