Which is the correct one?
- John neither bought a car nor a bike
- John bought neither a car nor a bike
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One reason to put "neither" in front of the verb is if you are contrasting two different actions (two different verbs). For example,
John neither bought a car nor rented a bike.
Since your first sentence is not contrasting two different actions, it could be argued that it is not strictly correct.
Since you are contrasting two different objects, then it is appropriate to put "neither" immediately before those objects, like in your second sentence.
That said, both of your sentences will be understood to mean the same thing by a native speaker.