One of the decompositions mentioned in the Wikipedia page on matrix decompositions is the Algebraic Polar Decomposition. This factors a square complex matrix $M$ into $M = SQ$ where $S = S^T$ and $QQ^T = I$. This is similar to the usual polar decomposition, except that in the usual case $S = overline S^T$ and $Qoverline Q^T = I$.
In a 1990 paper by Kaplansky, it is shown that this decomposition exists if and only if $M^T M$ is similar to $M M^T$, and in particular this is true for all invertible matrices.
Does this decomposition have any known applications? I guess an obstacle to applications is that the complex orthogonal matrices don’t form a compact space.