This is just by an accidental couriosity:
In the near of a root $\rho$ of the Riemann's zeta - can there be a continuous line starting from $\zeta(\rho)=0$ to $\zeta(\rho + \varepsilon_j)=a_j + î b_j $ with always $a_j = b_j$ (or, equivalently, $\operatorname{arg}(\zeta(\rho+\varepsilon_j))=\pi/4$ for a connected set of $\varepsilon_j$) ?
When I tried to think about this, I imagined a circle with some radius $r=|\varepsilon|$ enclosing the root and I feel tempted to think there must occur an equality somewhere on the circle -and thus by expanding continuously the radius of the circle from zero to some extent creating a continuous curve- but I'm sceptic about this and there might exist a real easy/obvious argument against it.
[update 2] By my own exercises it seems likely that such continuous lines exist. Now my question should be extended for the case, that this is indeed true: How would I formalize the argument, that there is indeed such a continuous line?
By a simple binary search I found for instance for a $|\varepsilon| = 0.01$ the value
$ \qquad \qquad \varepsilon \approx 0.00807689199924 + 0.00589608477150 î \qquad $ giving
$ \qquad \qquad \zeta(\rho_0 + \varepsilon) \approx 0.00559286683116 + 0.00559286683116 î $
and for a $|\varepsilon| = 1$ the value
$ \qquad \qquad \varepsilon \approx 0.548969374675 + 0.835842464624 î \qquad $ giving
$ \qquad \qquad \zeta(\rho_0 + \varepsilon) \approx 0.468005713605 + 0.468005713605 î $
[update 1]
Here is a picture of the (interpolated) locus of the equality for $\operatorname{arg}(\zeta(\rho_0+\varepsilon))=\pi/4$ where $\rho_0$ is the first nontrivial root of the zeta:
and here the detail of $\varepsilon$ alone:
[update 3] This image shows three "equi-tangent"-zeta curves, but where I do not yet know how they proceed furtherly to the left:
$arg(\zeta(x))=1/8 \pi , \qquad arg(\zeta(x))=2/8 \pi , \qquad arg(\zeta(x))=3/8 \pi$
I find this image especially interesting because it suggests that it might be unreasonable to assume two nontrivial roots on a short distance besides the vertical line z=1/2 with the same imaginary component , because if we have thus two nearby blue curves, then the magenta line of the one root and the green line of the other root should cross each other but which would then give a contradictory description for that matching point (but of course I'm not going to try to formalize this hypothesis) ...