A challenge with radial-velocity (RV) data is disentangling the origin of signals either due to a planetary companion or to stellar activity. Obtaining data over a large time span and wavelength range for this reason is crucial, but can oftentimes be difficult to achieve. The existence of a planetary companion has been proposed, as well as contested, around the relatively bright, nearby M3.0V star AD Leo at the same period as the stellar rotation of 2.23d. We further investigate the nature of this signal for the flare star. We introduce new CARMENES optical and near-IR RV data and an analysis in combination with archival data taken by HIRES and HARPS, along with more recent data taken by HARPS-N, GIANO-B, and HPF. Additionally, we address the confusion concerning the binarity of AD Leo. We consider possible correlations that the RVs may have with an assortment of stellar activity indicators accessible with CARMENES. We additionally performed a variety of modeling techniques considering all RV data using the Bayesian log evidence to determine whether a Keplerian planetary model, a red-noise quasi-periodic stellar activity model using a Gaussian process, or a mixed model would explain the observed data best. In addition, we take a closer look at spectral lines potentially associated with stellar activity. The CARMENES RV data agree with the previously reported periodicity of 2.23d, correlate with some activity indicators, and exhibit chromaticity. However, when considering the entire, extensive RV data set, we find that a mixed model composed of a stable and a variable component performs best. Moreover, when recomputing the RVs using only spectral lines insensitive to activity, there appears to be some residual power at the period of interest. We therefore conclude that it is not possible to determinedly prove that there is no planet orbiting in synchronization with the stellar rotation given our data, current tools, machinery, and knowledge of how stellar activity affects RVs. We do rule out planets more massive than 27M_{Earth} (=0.084M{Jup}). In addition, we rule out any binary companion around AD Leo with Msini greater than 3-6M{Jup}_ on orbital periods <14yr.