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Can someone please give me a reference(or a sketch of a proof), where I can find a proof that any algebraic extension of a complete field is complete ?

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This is only true for finite algebraic extensions. –  Julian Rosen Dec 4 '12 at 21:24
    
I read here that a counterexample is the algebraic closure of p-adic numbers. However, I won't have access to the proof until tomorrow. Does anyone know other counterexamples? By the way, in the finite extension case, it's quite easy and it relies ultimately on the fact that over a complete field all the norms over a finite dimensional vector space are equivalent. A proof can be found in Cassels, Frolich, Algebraic Number Theory, probably chapter I (or II). –  Mauro Porta Dec 4 '12 at 21:37
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