Consider the complex number z defined as:
$$ z = 3 + 4i $$
where i is the imaginary unit. Calculate the modulus and argument of the complex number w given by:
$$ w = z^2 + 4z + 8 $$
Next, express w in the form of a + bi, where a and b are real numbers. Finally, find the value of the argument of w rounded to three decimal places.