We show that the viscous Burgers equation $u_t+uu_x=u_{xx}$
considered for complex valued functions $u$ develops finite-time
singularities from compactly supported smooth data. By means of
the Cole-Hopf transformation, the singularities of $u$ are related
to zeros of complex-valued  solutions $v$ of the heat equation
$v_t=v_{xx}$. We prove that such zeros are isolated if they are
not present in the initial data.