Regular expression that have the same number of 0 and 1
Your $\alpha$ matches all strings, and is certainly incorrect.

The language is a union of two sets, so you can consider the two parts respectively and construct the regular expression with the form $R_1\mid R_2$.

The $\Sigma^*\{00,11\}\Sigma^*$ part is easy.

The $|w|_0=|w|_1$ part is somewhat tricky. You cannot construct a regular expression for $|w|_0=|w|_1$ because this language is not regular. However, You can only care about the set $S=\{w\mid |w|_0=|w|_1\text{but }w\notin\Sigma^*\{00,11\}\Sigma^*\}$ instead of the set $\{w\mid |w|_0=|w|_1\}$. Now try to figure out what $S$ looks like and write a regular expression for it. This is not hard.