Of or relating to alchemy; having the qualities or characteristics of alchemical practices and beliefs.
From alchemistic + -al suffix, creating an alternative, more elaborate form. This redundant double suffix (-ic + -al) was common in Early Modern English but is now archaic.
English once loved stacking suffixes—'alchemistical' sounds grander than 'alchemistic,' which is why Shakespeare and his contemporaries would have used the longer form. It's like the difference between 'magic' and 'magical' and 'magicality.'
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