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 ≳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 α profile appearing at ∼30 d post-maximum is the most atypical. The H α is at first characterised by two emission components, one at ∼+3000 km s −1 and a second at ∼ − 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  ∼ 2000–6000 km s −1 ), but the red component is significantly broader ( FWHM  ≳ 10 000 km s −1 ) and Lorentzian. The blue H α component evolves towards a lower-velocity offset before abruptly fading at ∼ + 100 d post-maximum brightness, concurrently with a light curve break. Multi-component profiles are observed in other hydrogen lines, including Pa β , 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  ∼ 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 α , we can constrain the distance of the CSM to be less than ∼6.5 × 10 15 cm (430 AU), or even lower (≲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.