Complement of equality problem of Turing machine is recognisable or not
Let's start with the definition of recognizable:

A language $L$ is recognizable if there exists a Turing machine $T$ such that if $x \in L$ then $T$ halts on $x$, and if $x \notin L$ then $T$ does not halt on $x$.

You haven't specified what you mean by &quot;complement of equality problem of Turing machines&quot;, so let me assume that you mean the language $L$ consisting of pairs $(T_1,T_2)$ of Turing machines which do not recognize the same language, that is, there exists $x$ such that $T_1$ halts on $x$ and $T_2$ doesn't, or vice versa.
You could try recognizing $L$ by going over all words $x$, and for which one, checking whether $T_1$ halts on $x$ and $T_2$ doesn't, or vice versa. While you can recognize that $T_1$ halts on $x$, you cannot recognize that $T_2$ doesn't halt on $x$, so this strategy fails.
To show that this language is unrecognizable, you can use a reduction from the non-halting problem. Suppose that $T$ is a Turing machine, and we want to determine whether it doesn't halt on the empty input. Construct a new Turing machine $T_1$ which erases its input and transfers control to $T$, and let $T_2$ be a fixed Turing machine which immediately halts. Then $(T_1,T_2) \in L$ iff $T$ does not halt on the empty input.