When we derive the expression for particle flux/probability current, we find that it is directly proportional to the Wronskian of $\psi$ and its conjugate and deduce that it is constant when particle current is constant.
Is it just a coincidence that the Wronskian shows up, or does the have some noteworthy implications?
Stick to one dimension, as the identification only holds for one space dimension, so you might well call it a coincidence, $$ j={\hbar\over 2mi}(\psi^*\partial_x \psi -\psi \partial_x \psi^*), \\ \partial_t \rho + \partial_x j =0 ~~~~\leadsto ~~~ v(x) = j/\rho ~~. $$
Vanishing Wronskian suggests (not quite dictates) that $\psi^*$ is linearly dependent on $\psi$, i.e. $\partial_x (\ln (\psi^*/\psi))=0$; so they may only differ by a constant phase, absorbable, hence the wave function is basically real, and the probability density constant over time. This is what you get in stationary systems.
A constant current J also indicates, for $\psi\equiv \sqrt{\rho} ~ e^{is(x)}$, a special velocity $$ J={\hbar\rho \over m} \partial_x s, ~~~~\leadsto ~~~ \psi\equiv \sqrt{\rho} ~~e^{i(\alpha+ \int^x \!dx' ~ Jm/\hbar \rho(x') ) } $$ so ρ is again time independent, even though ψ is genuinely complex.
