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Depends on the face. Infograph face, which I believe has the most, comes in at eight complications. Then you got the Hermes Circuit H with a whopping zero.

Haven’t noticed any slow downs while using the Infograph with eight complications but I’m sure there will be some battery drain with dynamic complications.
 
More complications won't slow down your watch. Depending on the complication(s) used, you could see more battery usage. If you have your watch set up to constantly notify you of every little thing, your battery will take a hit over the course of the day, even without using a complication. Sometimes, less complication usage can be much better on the watch and your mindset.
 
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