This is the sort of philosophical debate that doesn't have a clear right answer. I'll put my view on it out there, for what it's worth.
(All of what follows assumes a real variable. The complex logarithm will not appear.)
On $(0,\infty)$, $\ln x+C$ is an antiderivative for $\frac1x$. On $(-\infty,0)$, $\ln(-x)+C$ is an antiderivative for $\frac1x$. We can write these down in one formula as $\ln |x|+C$ - but that's deceptive. It creates the false impression that we can integrate across the singularity and get a value for something like $\int_{-2}^3 \frac1x\,dx$ by taking a difference in that antiderivative formula, when in fact that integral diverges.
In any convergent integral, we won't be crossing that singularity; any integral we can actually evaluate is entirely on one side. However, it's possible that our integral is on the negative side, so we want a general antiderivative that gives us options there. As such, here's my preferred form for the most general antiderivative of $\frac1x$: $\ln(Ax)$ for nonzero $A$. That's an antiderivative on the positive side if $A>0$, and it's an antiderivative on the negative side if $A<0$.