This won't be much of a "historical" answer (although my external links probably have interesting history in them), but hopefully it'll be mathematically useful, at least if construed as a long comment. I'll use Einstein notation throughout, except in blue expressions.
Consider a system of arithmetic on a finite-dimensional vector space over $\Bbb R$ (or your favourite subfield, e.g. $\Bbb Q$) with basis $e_i$ satisfying $a_ie_i+b_ie_i=(a_i+b_i)e_i$ and $a_ie_ib_je_j=u_{ijk}a_ib_je_k$ for coefficients $u_{ijk}$. We can write this as $ab=c$ with $c_k=u_{ijk}a_ib_j$. Axis $l$ exhibits the symmetry you describe iff $a_i^\prime=\color{blue}{(1-2\delta_{il})a_l}$ satisfies $(ab)^\prime=a^\prime b^\prime$, i.e.$$\color{blue}{[(1-2\delta_{il})(1-2\delta_{jl})-1+2\delta_{kl}])u_{ijk}}=0.$$You've noted the $u_{ijk}$ for complex numbers satisfy this condition for $l=1$ but not $l=0$, where $e_0:=1,\,e_1:=i$. @Conifold's point is $u_{ijk}=\color{blue}{\delta_{ij}\delta_{ik}}$ satisfies the condition for all $l$.
I also know that in physical space we also have one anisotropic axis (time) and three symmetric spatial axes.
I may not be able to walk into the past, but the equations describing nature are (largely) time-symmetric. (If that seems like a contradiction, bear in mind this is a big topic, sadly not addressed by the line of analysis you tried).
a question emerged, whether the real axis is naturally linked to the time dimension
The real mathematical reason time is dissimilar to space is the signs of real eigenvalues of the metric tensor. (There are two conventions regarding which sign time gets; in the linked article, the chosen convention makes time's eigenvalue negative, but I'll use the opposite convention hereafter in explaining the relevance.) Infinitesimal paths have squared length $ds^2=dt^2-dx^2-dy^2-dz^2$ in special relativity, if we use units where $c=1$, so $\sqrt{ds^2}$ is real for paths that don't exceed the speed of light, i.e. are causal. In more general geometries than Minkowski, the details become more complicated, but the signs of metric tensor eigenvalues are unchanged, and effectively allow us to count time and space dimensions.
Is the existence of no more than one asymmetric axis determined by the laws of mathematics?
As noted above, no. As for multiple time dimensions, that's also doable, albeit for unrelated reasons you can guess it in light of the paragraph above.