If $\mathbb{K}$ is any field and is endowed with the discrete topology, then $\mathbb{K}$ is a local field (*). further, if $\mathbb{K}$ is connected, then $\mathbb{K}$ is either $\mathbb{R}$ or $\mathbb{C}$ (**).
I was able to prove (*), but not (**).
edited: I am reading an article and above sentences are of the following paragraph of the article: Let $\mathbb{K}$ be a field and a topological space. Then $\mathbb{K}$ is called a local field if both $\mathbb{K}^+$ and $\mathbb{K}^*$ are locally compact Abelian groups, where $\mathbb{K}^+$ and $\mathbb{K}^*$ denote the additive and multiplicative groups of $\mathbb{K}$, respectively. If $\mathbb{K}$ is any field and is endowed with the discrete topology, then $\mathbb{K}$ is a local field. Further, if $\mathbb{K}$ is connected, then $\mathbb{K}$ is either $\mathbb{R}$ or $\mathbb{C}$. If $\mathbb{K}$ is not connected, then it is totally disconnected. Hence by a local field, we mean a field $\mathbb{K}$ which is locally compact, non-discrete and totally disconnected. The p-adic fields are examples of local fields.
For (*), I'm surprised by the definition you cite: a local field is usually defined as "a topological field with respect to a locally compact nondiscrete topology (with some extra properties)" (http://en.wikipedia.org/wiki/Local_field). If we don't require indiscretion, then your statement is trivially true since discrete topologies are locally compact; otherwise your statement is even more trivially true since there are no discrete local fields at all.
I'm interpreting (**) as being, "If $K$ is any connected local field, then $K$ is either $\mathbb{R}$ or $\mathbb{C}$." As Qiaochu points out, this statement requires that local compactness be part of the definition of local field (that is, it's false for merely topological fields). Under this assumption, the statement is true; see Are $\Bbb R$ and $\Bbb C$ the only connected, locally compact fields?.