The ability to visualize small moving objects is vital for the survival of many animals, as these could represent predators or prey. For example, predatory insects, including dragonflies, robber flies and killer flies, perform elegant, high-speed pursuits of both biological and artificial targets. Many non-predatory insects, including male hoverflies and blowflies, also pursue targets during territorial or courtship interactions. To date, most hoverfly pursuits have been studied outdoors. To investigate hoverfly (Eristalis tenax) pursuits under more controlled settings, we constructed an indoor arena that was large enough to encourage naturalistic behavior. We presented artificial beads of different sizes, moving at different speeds, and filmed pursuits with two cameras, allowing subsequent 3D reconstruction of the hoverfly and bead position as a function of time. We show that male E. tenax hoverflies are unlikely to use strict heuristic rules based on angular size or speed to determine when to start pursuit, at least in our indoor setting. We found that hoverflies pursued faster beads when the trajectory involved flying downwards towards the bead. Furthermore, we show that target pursuit behavior can be broken down into two stages. In the first stage, the hoverfly attempts to rapidly decreases the distance to the target by intercepting it at high speed. During the second stage, the hoverfly's forward speed is correlated with the speed of the bead, so that the hoverfly remains close, but without catching it. This may be similar to dragonfly shadowing behavior, previously coined 'motion camouflage'.