Living fluids, arising from a complex organization of matter driven at the scale of its constituent agents, are confounding, and can exhibit what is known as "active turbulence". Analogies with high Reynolds number, and inertial (Kolmogorov) turbulence, however, have not survived beyond the qualitative. The lack of scale separation in low Reynolds active flows breaks away from the familiar notions of the energy cascade and approximate scale-invariance of inertial turbulence. So do experimental studies hinting that active turbulence can manifest anomalous diffusion, while inertial turbulence does not surpass classical diffusion. Using a continuum hydrodynamic model for dense bacterial suspensions, we investigate the Lagrangian and Eulerian features distinguishing active from inertial turbulence. We find a robust flow transition to universality beyond a critical activity, which is accompanied by intermittency and maximal chaos. Interestingly, this asymptotic flow state manifests super diffusion via Lévy walks, and consequently Lagrangian anomalies, which we trace to oscillatory streaks emerging in the flow field. All of this makes the phenomenology of living matter flows rich and riddled with surprising nuances, which at times bridge the analogy with inertial turbulence, and at others break them. Broad-brushed parallels, therefore, obfuscate what may be biologically relevant strategies for survival and growth.