ABSTRACT We present a detailed study of the complex time-frequency structure of a sample of previously reported bursts of FRB 121102 detected with the MeerKAT telescope in September 2019. The wide contiguous bandwidth of these observations have revealed a complex bifurcating structure in some bursts at 1250 MHz. When de-dispersed to their structure-optimized dispersion measures (DMs), two of the bursts show a clear deviation from the cold plasma dispersion relationship below 1250 MHz. We find a differential DM of ${∼ }1{-}2~{\rm pc \, cm^{-3}}$ between the lower and higher frequency regions of each burst. We investigate the possibility of plasma lensing by Gaussian lenses of ∼10 au in the host galaxy, and demonstrate that they can qualitatively produce some of the observed burst morphologies. Other possible causes for the observed frequency dependence, such as Faraday delay, are also discussed. Unresolved sub-components in the bursts, however, may have led to an incorrect DM determination. We hence advise exercising caution when considering bursts in isolation. We analyse the presence of two apparent burst pairs. One of these pairs is a potential example of upward frequency drift. The possibility that burst pairs are echoes is also discussed. The average structure-optimized DM is found to be $563.5± 0.2 (\text{sys}) ± 0.8 (\text{stat})\, {\rm pc \, cm^{-3}}$ – consistent with the values reported in 2018. We use two independent methods to determine the structure-optimized DM of the bursts: the DM_phase algorithm and autocorrelation functions. The latter – originally developed for pulsar analysis – is applied to fast radio bursts for the first time in this paper.