PURPOSE: To produce a new complex suitable to an α-olefin polymerization catalyst by allowing each of specified reactants to react with anhydrous hydrogen halide in the presence of magnesium halide of alcohol and in a hydrocarbon reaction medium without containing ether.

CONSTITUTION: An objective complex is obtained by allowing a reactant selected from a group consisting of a magnesium metal, dialkyl magnesium, alkyl magnesium halide, alkyl magnesium alkoxide, magnesium dialkoxide and alkoxy magnesium halide to react with anhydrous hydrogen halide in the presence of ≤6 mol per mol magnesium halide of alcohol selected from chloro-substd. 1-10C alcohol and a mixture of chloro-substd. alcohol and β-alkyl-substd. alcohol and in a hydrocarbon reaction medium without containing ether.