Is "hasta semana" correct informal Spanish for "see you next week?
Or would you say something more elaborate such as, "hasta la próxima semana"?

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Se dice "Hasta la próxima semana" o "te veo la semana que viene"

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hello, you can say "te veo hasta la otra semana" as a formal way but you can say it in an informal way, "nos vemos la otra semana"

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Te veo la próxima semana.

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hello!!
You can use:
"Hasta la siguiente semana" 
"Hasta la proxima semana" 
"Hasta la otra semana"
"Hasta la semana que viene"