The coherent way of extending the polynomial hierarchy is not indexed by ordinals, but by functions $a\colon\mathbb N\to\mathbb N$:
$$\Sigma_{a(n)}\mathrm P=\bigcup_{c\in\mathbb N}\Sigma_{a(n)}\text-\mathrm{TIME}(n^c)$$
is the class of problems computable by a polynomial-time alternating Turing machine that starts in an existential state and makes at most $a(n)-1$ alternations, where $n$ is the length of the input. (This agrees with the usual definition of the polynomial hierarchy when $a$ is a constant.)
When $a$ is an increasing function, one additional alternation hardly matters, while the class $\Sigma_{a(n)}\mathrm P$ will be quite fragile as any kind of operation will likely increase $a$. For this matter, in this context you are more likely to find more robust classes defined by a set of $a$ of certain growth, such as $\Sigma_{O(\log n)}\mathrm P$. Then it is no longer relevant what state you start with, so such classes are more commonly denoted in the literature by something like
$$\mathrm{TimeAlternations}(t(n),a(n)),$$
e.g., $\mathrm{TimeAlternations}(n^{O(1)},O(\log n))$, where the resource bound now simply counts the number of alternations.
Unrestricted alternating polynomial time $\mathrm{AP}=\Sigma_\infty\mathrm P=\Sigma_{n^{O(1)}}\mathrm P$ equals PSPACE, so that’s where it stops.
These classes are considered rather rarely, as there are next to no natural situations where they arise. Even inside the usual polynomial hierarchy, you hardly ever encounter problems outside, say, $\Sigma_2^P\cup\Pi_2^P$. There are many interesting complexity classes between PH and PSPACE, but they are defined in different ways than by bounding the number of alternations: typically, they are variants of counting classes, where the decision is made based on the number of accepting paths of the Turing machine. See the basic counting classes PP (wp, zoo) and #P (wp, zoo; this is a class of function problems rather than decision problems), the modular counting classes $\oplus\mathrm P$ (wp, zoo) and more generally $\mathrm{Mod}_p\mathrm P$ (zoo), the counting hierarchy CH (wp, zoo) and modular counting hierarchy ModPH (zoo) (for a fixed prime $p$, the more basic hierarchy $\mathrm{Mod}_p\mathrm{PH}$ collapses to $\mathrm{BP\cdot Mod}_p\mathrm P$ by Toda’s theorem), and various related classes such as MP (zoo).
Anyway, if you try to define a hierarchy indexed by ordinals, it’s quite unclear what to do already at the $\omega$ level. To make it a “limit” of the constant-$k$ $\Sigma_k\mathrm P$ classes, it should allow an unbounded number of quantifiers/alternations, i.e., it should be something like $\Sigma_{a(n)}\mathrm P$ for a suitable function (or a set of functions) $a$, but which one? Different choices yield different classes. (Arguably, the most natural choice, which corresponds to a most obvious way of “diagonalizing” the PH levels, would be to allow polynomial number of alternations, in which case the $\omega$th level is PSPACE and the hierarchy stops.) Furthermore, whatever choice you take, level $\omega+1$ should add one more existential quantifier in front, which just increases $a(n)$ to $a(n)+1$. If you define level $\omega$ using a robust set of functions rather than a single one, this might simply be the same class; either way, it is a class of the same kind as level $\omega$, extept for the, quite arbitrary, choice of the function $a$. If you try to continue to define an ordinal-indexed hierarchy in this way, you will end up with something like $\Sigma_{f_\alpha(n)}\mathrm P$ for an ordinal-indexed family of functions $f_\alpha(n)$, for which there are no natural or canonical choices. In the end, the only robust way to do this is to abandon ordinals and just consider the classes $\Sigma_{a(n)}\mathrm P$ directly.
