Asymptotic Complexity of Gaussian Elimination using Complete Pivoting

I would like to know the algorithm asymptotic complexity with Complete Pivoting. With partial pivoting, it is known to be $$O(n^3)$$.

Is it the same for complete pivoting?

Yes. Searching the entire trailing submatrix for the next pivot instead of only the current column merely replaces the time spent on finding pivots from $$O(n^2)$$ to $$O(n^3)$$. While the total runtime is increased, the overall asymptotic complexity remains $$O(n^3)$$.
Carl's answer is correct, I upvoted it too. The growth in pivot searches from taking $$O(n^2)$$ steps to $$O(n^3)$$ steps is unfortunate, but doesn't jeopardize the overall complexity. But I think the usual caveats about big-$$O$$ notation should be doubly repeated, that omitting the constants can obscure important details.