We consider a family of free multiplicative Brownian motions $b_{s,tau}$ parametrized by a positive real number $s$ and a nonzero complex number $tau$ satisfying $leftvert tau-srightvert leq s,$ with an arbitrary unitary initial condition. We compute the Brown measure $mu_{s,tau}$ of $b_{s,tau}$ and find that it has a simple structure, with a density in logarithmic coordinates that is constant in the $tau$-direction. We also find that all the Brown measures with $s$ fixed and $tau$ varying are related by pushforward under a natural family of maps. Our results generalize those of Driver-Hall-Kemp and Ho-Zhong for the case $tau=s.$ We use a version of the PDE method introduced by Driver-Hall-Kemp, but with some significant technical differences.
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