If there is a subject in the complement, the verb in the complement is finite (tensed):
The little girl hopes {her mom buys her a doll}.
and when there is no subject in the complement, the verb in the complement is an infinitive:
The mother hopes {to buy her daughter a doll}.
Thus:
I hope {you buy me a doll}.
The second-person singular and second-person plural form of the verb buy is also buy, so the rule is not as easy to see as it is in the third person.
P.S.
Hope is complemented by a that-clause or reduced that-clause or an infinitive-phrase:
The little girl hopes {that her mom buys her a doll}.
The little girl hopes {__ her mom buys her a doll}.
The mother hopes {to buy her daughter a doll}.
want is completed with an infinitive phrase and with something else. It gets very convoluted, and I would need to use my lifeline, and place a virtual phone call to @StoneyB or @Araucaria.
The little girl wants {her mom to buy her a doll}
The little girl wants {that her mom should buy her a doll}. marginal
The mother wants {to buy her daughter a doll}.
P.P.S.
want can take a direct object; hope cannot:
He wants snow. He likes to ski.
not OK He hopes snow.
OK He hopes for snow.
P.P.P.S.
The little girl wants {her mom to buy her a doll}
The little girl hopes {her mom buys her a doll}.
Why is "her mom" a subject with hopes but not a subject with wants?
I think it is because the underlying idea of want is to lack<object> and the underlying idea of hope is to await<event>.