We consider a family of free multiplicative Brownian motions $b_{s,τ}$ parametrized by a real variance parameter $s$ and a complex covariance parameter $τ.$ We compute the Brown measure $μ_{s,τ}$ of $ub_{s,τ},$ where $u$ is a unitary element freely independent of $b_{s,τ}.$ We find that $μ_{s,τ}$ has a simple structure, with a density in logarithmic coordinates that is constant in the $τ$-direction. These results generalize those of Driver-Hall-Kemp and Ho-Zhong for the case $τ=s.$ We also establish a remarkable "model deformation phenomenon," stating that all the Brown measures with $s$ fixed and $τ$ varying are related by push-forward under a natural family of maps. Our proofs use a first-order nonlinear PDE of Hamilton-Jacobi type satisfied by the regularized log potential of the Brown measures. Although this approach is inspired by the PDE method introduced by Driver-Hall-Kemp, our methods are substantially different at both the technical and conceptual level.