Division by imaginary number

The incorrect step is saying:

$\sqrt{4}/\sqrt{-1} = \sqrt{4/-1}$

The identity:

$\sqrt{a}/\sqrt{b} = \sqrt{a/b}$

is only justified when $a$ and $b$ are positive.

The only soundproof way to be sure to find the right result while dividing two complex numbers

$$\frac{a+bi}{c+di}$$

is reducing it to a multiplication. The answer is of the form $x+yi$; therefore

$$(c+di)(x+yi) = a+bi$$

and you will end up with two linear equations, one for the real coefficient and another for the imaginary one. As Simon and Casebash already wrote, taking a square root leads to problems, since you cannot be sure which value must be chosen.