Lipid bilayers are quasi two-dimensional, crowded systems consisting of phospholipid molecules and membrane proteins. It is known that the lipids and proteins undergo thermally driven lateral diffusion and thus constantly reorganize the membrane. Using extensive simulation and theoretical modelling of anomalous diffusion, we quantify the stochastic properties of the lateral diffusion in membranes at varying circumstances. We show that the lateral diffusion gets more anomalous characters as the membrane complexity is increased by adding cholesterol molecules or proteins. In the absence of membrane proteins the anomalous motion is universally described by the gaussian anomalous diffusive process called fractional Brownian motion (FBM) regardless of molecular details of the lipid structure and lipid phases. However, it is found that this FBM picture is no longer valid if the membrane is crowded with proteins. We present how the crowding of membrane proteins changes statistical complexities of the anomalous diffusion in membranes.