Has the notion of having a complex amount of dimensions ever been described? And what about negative dimensionality?

Negative dimension is actually much easier to talk about than complex dimension. Super vector spaces are a natural collection of objects that can have negative dimension; given a super vector space $(V_0, V_1)$ we can define its dimension to be $\dim V_0 - \dim V_1$, and this definition has many nice properties; see this blog post, for example.

More generally, there is a natural notion of dimension in any (braided?) monoidal category with duals (see Traces in symmetric monoidal categories by Ponto and Shulman for a definition and thorough discussion). It includes as special case many notions of Euler characteristic, and in particular is frequently negative, although it is not always a number; in general it takes values in the monoid $\text{End}(I)$ where $I$ is the identity object. (If the category is preadditive with the monoidal product distributing over addition of morphisms, then $\text{End}(I)$ is a ring, and one can ask whether it is isomorphic to a subring of $\mathbb{C}$.)

Algebraic stacks are a far-reaching generalization of algebraic varieties. If an algebraic variety is considered as a stack, then its dimension as stack is the same as its dimension as variety. However there are many stacks that do not correspond to varieties, and some of these have negative dimension.

Specifically, if $V$ is a variety and $G$ is an algebraic group acting on $V$, then we can always form the quotient stack $[V/G]$ (which in most cases won't be a variety). Then we have $$\dim([V/G])=\dim(V)-\dim(G)$$ which may well be negative. For instance, if you let $G$ act trivially on a point $P$, then the quotient stack $[P/G]$, known as the classifying stack of $G$, will have dimension $\dim([P/G])=-\dim(G)$.

The same game can be played with differentiable stacks, smooth manifolds, and Lie groups acting on those manifolds. Also: topological stacks, topological manifolds, and topological groups acting on those manifolds.

One notion of complex dimension that has been used extensively has to do with self-similar sets. A $t$-neighborhood (i.e. points within distance $t$) of such a set may have volume $v(t)$ bounded above and below by constant multiples of $t^d$, where $d$ is the dimension of the boundary and $t$ is small, but such that $t^{-d} v(t)$ is oscillatory and non-convergent as $t$ goes to zero. In such a case the oscillatory information may sometimes be described using a complex power of $t$. If $v(t)=t^{d+ci}= t^d \exp(ic \log t)$ one could think of the dimension being $d+ci$. In general sets will have many complex dimensions. There are applications to Weyl asymptotics and connections to number theory (spectral zeta functions).

For details see the work of Michel Lapidus (on fractal strings) or the paper of Erin Pearse on complex dimensions of self-similar systems.