Carbon nanotubes (CNTs) based composites such as carbon quantum dot-CNT (CQD-CNT) demonstrate excellent performance in the fields of bioimaging and biomedicine. Nevertheless, CQD-CNT composites were mostly prepared by multiple and cumbersome steps, including the synthesis of CQDs, functionalization of CNTs and subsequent assembly of CQDs on CNTs. Here, a facile one-step synthesis of CQD-CNT was successfully conducted on an active CaO supported FeCo (Fe-Co/CaO) catalyst derived from waste eggshells via chemical vapor deposition, and other CNT-based composites such as carbon fiber-graphene sheet-CNT (CNF-GNS-CNT) and CNF-CNT as well as CNTs were also synthesized by modulating the catalyst support, growth temperature and growth time. The catalysts and produced CNTs and CNT-based composites were analyzed by multiple characterization tools. The results showed that nano CaO supported FeCo catalyst only synthesized MWCNTs, whereas Fe-Co/CaO was favorable for the synthesis of CNT-based composites. Cobalt oxides and iron oxides were well dispersed in the fresh Fe-Co/CaO catalyst, and FeCo alloy particles of different sizes were formed after catalyst reduction and heating to different growth temperatures. At a low growth temperature of 750 °C, CNF-GNS-CNT was synthesized on Fe-Co/CaO, in which CNTs and CNFs were grown on small (~10 nm) and larger (~30 nm) metal particles, respectively, whereas GNSs were mainly produced on CaO. At a high growth temperature of 850 °C, surface area and pore volume of catalysts decreased greatly, and higher solubility of carbon atoms in metal particles resulted in less desorbed CHx, which is not favorable for the growth of GNSs. Instead, CQD-CNT composites were selectively synthesized at 850 °C, which was ascribed to the oxidation corrosion of CNFs by OH radicals, but with the extension of growth time, only CNTs were produced due to the complete oxidation of CQDs. Further, larger metal particles were produced at a higher temperature of 950 °C, which were wrapped by graphite, and the partial deactivation of metal particles led to a few small particles remained active for CNT growth with a low carbon yield.