In circumstellar gas, the complex organic molecule methanol has been found almost exclusively around young stellar objects, and is thus regarded as a signpost of recent star formation. Here we report the first probable detection of methanol around an evolved high-mass star, in the complex circumstellar environment around the Luminous Blue Variable $eta$ Carinae, while using ALMA to investigate molecular cloud conditions traced by CO (2-1) in an orbit phase of the massive binary preceding the 2020 periastron. Favoring methanol over a $^{13}$CS alternative, the emission originates from hot ($T_{rm{gas}} simeq$ 700 K) material, $sim$2$$ (0.02 pc) across, centered on the dust-obscured binary in contrast to the CO which traces inner layers of the extended massive equatorial torus, and is accompanied by prominent absorption in a cooler ($T_{rm{gas}} simeq$ 110 K) layer of gas. We also report detections of water in $Herschel$/HIFI observations at 557 GHz and 988 GHz. The methanol abundance is several to 50 times higher than observed towards several lower mass stars, while water abundances are similar to those observed in cool, dense molecular clouds. The very high methanol:water abundance ratio in the core of $eta$ Carinae may suggest methanol formation processes similar to Fischer-Tropsch-type catalytic reactions on dust grains. These observations prove that complex molecule formation can occur in the chemically evolved environments around massive stars in the end stages of their evolution, given sufficient gas densities and shielding conditions as may occur in material around massive interacting companions and merger remnants.
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