The free multiplicative Brownian motion $b_{t}$ is the large-$N$ limit of Brownian motion $B_t^N$ on the general linear group $\mathrm{GL}(N;\mathbb{C})$. We prove that the Brown measure for $b_{t}$---which is an analog of the empirical eigenvalue distribution for matrices---is supported on the closure of a certain domain $Σ_{t}$ in the plane. The domain $Σ_t$ was introduced by Biane in the context of the large-$N$ limit of the Segal--Bargmann transform associated to $\mathrm{GL}(N;\mathbb{C})$. We also consider a two-parameter version, $b_{s,t}$: the large-$N$ limit of a related family of diffusion processes on $\mathrm{GL}(N;\mathbb{C})$ introduced by the second author. We show that the Brown measure of $b_{s,t}$ is supported on the closure of a certain planar domain $Σ_{s,t}$, generalizing $Σ_t$, introduced by Ho. In the process, we introduce a new family of spectral domains related to any operator in a tracial von Neumann algebra: the {\em $L^p_n$-spectrum} for $n\in\mathbb{N}$ and $p\ge 1$, a subset of the ordinary spectrum defined relative to potentially-unbounded inverses. We show that, in general, the support of the Brown measure of an operator is contained in its $L_2^2$-spectrum.