For ammonium removal from wastewater, anammox technologies are among the most efficient and rapidly developing ones. Due to the low growth rate of anammox bacteria and their sensitivity to various inhibitors, technologies using attached biocenosis carriers (ABCs) provide for reliable operation. The goal of the present work was to investigate a new ABC type, ETEK biochips based on a nonwoven fibrous material. The work involved the techniques of materials science (design of a new ABC type) and physical modeling of the anammox process (in a laboratory bioreactor), as well as electron microscopy and molecular profiling of activated sludge communities. Comparison of the ETEK biochips with the ABCs of foamed polyethylene BF33 and Mutag revealed more rapid accumulation (5-fold) of the activated sludge biomass on ETEK biochips upon reactor launching, as well as comparable buoyancy and reactor productivity regarding N removal. The specific rate of nitrogen removal obtained with ETEK biochips considerably exceeded that for foamed polyethylene with a filler: 1.5–3 times higher per chip and 1.5 times higher per activated sludge biomass unit. The studied ABC shared the same issue of floating to the surface due to the active formation of gas (N2). The algorithm for calculating the downward flows in bioreactors with rapidly surfacing ABC is proposed, and a new hydrodynamic type of a bioreactor (with hybrid hydrodynamics) is described, a moving bed–sequencing batch reactor (MB-SBR).