Complexity of computing generalised determinants. (P - #P transition)
I extend my comment in an answer. 

By rewriting $e^{i \cdot sgn(\mu)\theta} = \cos(sgn(\mu)\theta)+i\sin(sgn(\mu)\theta) = \cos(\theta)+i \cdot sgn(\mu)\sin(\theta)$ we have: $Det_\theta(A) = \cos(\theta)Perm(A)+i\sin(\theta)Det(A)$. Thus, if $\cos(\theta) \neq 0$, we have $Perm(A) = (Det_\theta(A)-i\sin(\theta)Det(A))/\cos(\theta)$, meaning that $Det_\theta$ is $\#P$-hard (and $VNP$-complete over $\mathbb{C}$).

Several generalizations of the determinant have been studied however. Peter Bürgisser studies the complexity of immanants in his Habilitationschrift "Completeness and Reduction in Algebraic Complexity Theory" (http://math-www.uni-paderborn.de/agpb/work/habil.ps). Nicolas de Rugy-Altherre studied the complexity of one of them, the Fermionant, and proves other $VNP$-completeness results for such functions in "Determinant versus Permanent: salvation via generalization? The algebraic complexity of the Fermionant and the Immanant", https://arxiv.org/abs/1309.2156.

Hyperdeterminants (https://en.wikipedia.org/wiki/Hyperdeterminant) is a generalization of determinants to tensors. See "New algorithms for linear k-matroid intersection and matroid k-parity problems" by A. Barvinok for identities and algorithms on hyperdeterminants.