We present a spectroscopic analysis of the most nearby Type I superluminous supernova (SLSN-I), SN 2018bsz. The photometric evolution of SN 2018bsz has several surprising features, including an unusual pre-peak plateau and evidence for rapid formation of dust greater than or similar to 200 d post-peak. We show here that the spectroscopic and polarimetric properties of SN 2018bsz are also unique. While its spectroscopic evolution closely resembles SLSNe-I, with early O II absorption and C II P Cygni profiles followed by Ca, Mg, Fe, and other O features, a multi-component H alpha profile appearing at similar to 30 d post-maximum is the most atypical. The H alpha is at first characterised by two emission components, one at similar to+3000 km s(-1) and a second at similar to - 7500 km s(-1), with a third, near-zero-velocity component appearing after a delay. The blue and central components can be described by Gaussian profiles of intermediate width (FWHM similar to 2000-6000 km s(-1)), but the red component is significantly broader (FWHM greater than or similar to 10000 km s(-1)) and Lorentzian. The blue H alpha component evolves towards a lower-velocity offset before abruptly fading at similar to + 100 d post-maximum brightness, concurrently with a light curve break. Multi-component profiles are observed in other hydrogen lines, including Pa beta, and in lines of Ca II and He I. Spectropolarimetry obtained before (10.2 d) and after (38.4 d) the appearance of the H lines shows a large shift on the Stokes Q - U plane consistent with SN 2018bsz undergoing radical changes in its projected geometry. Assuming the supernova is almost unpolarised at 10.2 d, the continuum polarisation at 38.4 d reaches P similar to 1.8%, implying an aspherical configuration. We propose that the observed evolution of SN 2018bsz can be explained by highly aspherical, possibly disk-like, circumstellar material (CSM) with several emitting regions. After the supernova explosion, the CSM is quickly overtaken by the ejecta, but as the photosphere starts to recede, the different CSM regions re-emerge, producing the peculiar line profiles. Based on the first appearance of H alpha, we can constrain the distance of the CSM to be less than similar to 6.5 x 10(15) cm (430 AU), or even lower (less than or similar to 87 AU) if the pre-peak plateau is related to an eruption that created the CSM. The presence of CSM has been inferred previously for other SLSNe-I, both directly and indirectly. However, it is not clear whether the rare properties of SN 2018bsz can be generalised for SLSNe-I, for example in the context of pulsational pair instability, or whether they are the result of an uncommon evolutionary path, possibly involving a binary companion.