Let $U$, $V$ be the image and kernel of $P$, respectively. Then $\mathbb{C}^n=U\oplus V$ and
$$
\|P\|\leqslant C\,\, \text{for some}\,\,C>1\Leftrightarrow \forall u\in U, v\in V\colon\,\|u\|^2\leqslant C^2 \|u+v\|^2 \\
\Leftrightarrow \forall u\in U, v\in V\colon\,0\leqslant (C^2-1) \|u\|^2+2C^2 \Re \langle u,v\rangle+C^2 \|v^2\|\\
\forall u\in U, v\in V,t\in \mathbb{R}\colon 0\leqslant
(C^2-1) t^2\|u\|^2+2C^2 t\Re \langle u,v\rangle+C^2 \|v^2\|\\
\Leftrightarrow \forall u\in U, v\in V\colon\,C^4 \Re \langle u,v\rangle^2\leqslant C^2(C^2-1) \|v^2\|\cdot \|u\|^2,
$$
here $U$ and $V$ come in symmetric fashion, so $\|P\|\leqslant C$ if and only if $\|I-P\|\leqslant C$.

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