[earthicus]

As others have pointed out, it is certainly not turing complete. The spectacular technical achievement of the antikythera mechanism, apart from its overall complexity, is the presence of differential gearing. Specifically, a differential turntable allows it to add or subtract angular velocities: you subtract the effect of the suns movement from the lunar movement to compute the lunar cycle.

PS Gears and cog wheels were only invented a century or two before the antikythera mechanism; scientific progress during the Hellenistic era was very rapid! Then the decline after the roman conquest was quite rapid as well - by the imperial period (eg Heron's writings) techniques like differential gears had already been lost, and they wouldn't be reinvented till (i think?) the 18th century!