Not only do $\def\L{{\sf L}}\L^p$-spaces make sense for all complex $p$, but their noncommutative generalizations play a crucial role in the Tomita–Takesaki modular theory.
One construction is described in the answer to the question Is there an introduction to probability theory from a structuralist/categorical perspective?.
The input data is a measurable space $(X,M)$ together with a σ-ideal $N$ of negligible (alias measure 0) sets. This is less data than a traditional measure space, since $\L^p$-spaces do not depend on the choice of a measure, only on the σ-ideal of negligible sets.
Below, it is convenient to replace $p$ by its reciprocal $1/p$. Thus, what used to be $\L^2$ is now denoted by $\L^{1/2}$ and $\L^∞$ is denoted by $\L^0$.
Now the space $\L^p(X,M,N)$ can be constructed as the quotient of the set of pairs $(f,μ)$ by an equivalence relation $\sim$.
Here $f$ is an equivalence class of bounded measurable functions on $(X,M,N)$ (modulo equality almost everywhere), $μ$ is a measure on $(X,M,N)$ (required to vanish on elements of $N$), and $μ$ is required to be finite if $\Re p>0$.
The equivalence relation is generated by $$(fg^p,μ)\sim (f,gμ),$$
where $g$ is an equivalence class of a strictly positive real measurable function on $(X,M,N)$ and $gμ$ denotes the multiplication of a measure by a function, as in the Radon–Nikodym theorem.
Equivalently, if $\Re p>0$, we can drop the boundedness condition on $f$ and finiteness condition on $μ$ and instead require that the measure $|f|^{1/\Re p}μ$ is finite. This construction produces the same $\L^p$-space because
$$(f,μ)=(\arg f\cdot (|f|^{1/\Re p})^{-\Im p}(|f|^{1/\Re p})^p,μ)\sim(\arg f\cdot (|f|^{1/\Re p})^{-\Im p},|f|^{1/\Re p}μ),$$
and in the last pair, the first component has absolute value at most 1 (hence is bounded), whereas $|f|^{1/\Re p}μ$ is a finite measure.
Here and below we write $p=\Re p+\Im p$ for the real and purely imaginary parts of $p$.
The $\L^p$-norm in this description is simply the $\Re p$-th power of the integral of $|f|^{1/\Re p}μ$.
To see that the resulting $\L^p$-space is isomorphic to the traditional $\def\LL{{\rm L}}\LL^p$-space $$\LL^p(X,M,μ)=\left\{f\biggm| \int|f|^{1/p} dμ<∞\right\},$$
observe that for any faithful measure $μ$ the map
$$f↦(f,μ)$$
yields an isomorphism $$\LL^p(X,M,μ)→\L^p(X,M,N),$$
where the right side does not depend on the choice of $μ$.
As a special case,
$\L^0(X,M,N)$ is the commutative von Neumann
algebra of equivalence classes of bounded measurable functions on $(X,M,N)$,
whereas $\L^1(X,M,N)$ is the Banach space of finite complex-valued measures on $(X,M,N)$.
The condition $f≥0$ singles out the positive subspace $\L^p_{≥0}⊂\L^p$, and the map $$μ↦(1,μ),\qquad \L^1_{≥0}→\L^p_{≥0}$$
is a bijection.
Here $\L^1_{≥0}$ can be identified with finite (positive real) measures on $(X,M,N)$.
Furthermore, given a complex number $q$, we can define the power operation $$\L^p_{≥0}(X,M,N)→\L^{pq}(X,M,N),\qquad (1,μ)↦(1,μ),$$
using which one can show that $(f,μ)=fμ^p$, justifying the above construction and the equivalence relation $\sim$.
The Hölder inequality yields a canonical multiplication map
$$\L^p(X,M,N)⊗\L^q(X,M,N)→\L^{p+q}(X,M,N)$$
given by
$$((f,μ),(g,μ))↦(fg,μ),$$
where we used the equivalence relation $\sim$ to pick representatives for elements of $\L^p$ and $\L^q$ with the same measure $μ$ as the second component.
The induced map
$$\L^p(X,M,N)⊗_{\L^0(X,M,N)}\L^q(X,M,N)→\L^{p+q}(X,M,N)$$
is an isomorphism, where $⊗$ denotes the usual algebraic tensor product, which happens to be automatically complete.
Thus,
$$\L^p(X,M,N)≅\L^{\Re p}(X,M,N)⊗_{\L^0(X,M,N)}\L^{\Im p}(X,M,N),$$
so the study of $\L^p$-spaces for complex $p$ reduces to the case of real $p$, which is well-known, and the case of purely imaginary $p$.
For a purely imaginary $p$, given a choice of a faithful measure $μ$, the map $$\L^0(X,M,N)→\L^p(X,M,N),\qquad f↦fμ^p$$
is an isomorphism, with its inverse being the map $f↦fμ^{-p}$.
Thus, for traditional measure spaces, we do not see something radically new, although one can find some satisfaction in the fact that the above constructions of $\L^p$-spaces are independent of the choice of a measure, whereas the isomorphism $\L^0→\L^p$ requires such a noncanonical choice, so from a conceptual point of view, there is some value in $\L^p$-spaces for purely imaginary $p$.
All of the above generalizes to the case of noncommutative von Neumann algebras.
For a purely imaginary $p$,
the isomorphism $\L^0→\L^p$ is now only an isomorphism of left modules over a (noncommutative) von Neumann algebra.
Indeed, the bimodules $\L^0$ and $\L^p$ are not isomorphic for a type III factor.
This essentially the Tomita–Takesaki modular theory of von Neumann algebra, see the answer to Making sense of "every non-commutative algebra has its own internal time evolution (aka a one-parameter group)"? for more information.
