Let $V$ be a finite dimensional vector space over $\mathbb{C}((t))$. Let $D:V\rightarrow V$ be a differential operator; i.e., an additive $\mathbb{C}$-linear map satisfying $$ D(a.v)=(t\frac{d}{dt}a).v,\quad \quad a\in \mathbb{C}((t)), \quad v\in V. $$ The fundamental theorem of Turrittin and Levelt states that $D$ has a Jordan decomposition. In more details, after possibly extending the field by adding a root of $t$, there exists a basis of $V$ in which $D$ is represented by a Jordan matrix. A good reference here is Levelt's original paper on the subject.
In the past decade or so, several people have developed a "$p$-adic analogue" of this theorem. Some of the people involved are Mebkhout, Andre, and Kedlaya. Christol has an unfinished book on this topic. Sometimes the result is known as $p$-adic monodromy theorem.
I would like to understand the statement of this result which shows its resemblance to the usual Levelt-Turrittin theorem mentioned above. Presumably, one has to replace $\mathbb{C}((t))$ by another field, and then the result states that differential operators over this field also have Jordan decompositions. Unfortunately, most of the references on the subject are full of technical jargon and seem impossible to penetrate for a novice. So I appreciate any helpful thoughts are references on this question.
