I learned the following "construction" in the article "Equivalent complete norms and positivity." from Arendt and Nittka. On a Banach space $(X,\lVert\cdot\rVert)$, take an unbounded functional $\varphi$ and a point $y\in X$ such that $\varphi(y)=1$ and define the operator $S:X\to X$ by $Sx := x - 2\varphi(x)y$. Then you can check that $S^2=I$ and $\lvert x \rvert := \lVert Sx\rVert$ defines a complete norm on $X$ which is not equivalent to the $\lVert\cdot\rVert$-norm.

Of course there are many such norms, you might also have a look at this question.

I learned the following "construction" in the article "Equivalent complete norms and positivity." from Arendt and Nittka. On a Banach space $(X,\lVert\cdot\rVert)$, take an unbounded functional $\varphi$ and a point $y\in X$ such that $\varphi(y)=1$ and define the operator $S:X\to X$ by $Sx := x - 2\varphi(x)y$. Then you can check that $S^2=I$ and $\lvert x \rvert := \lVert Sx\rVert$ defines a complete norm on $X$ which is not equivalent to the $\lVert\cdot\rVert$-norm.

Of course there are many such norms, you might also have a look at this question.