Solution:

Write $w = x + yi$ with $x, y$ real. The sum of the roots is $4w + 12i - 4 = -a,$ which is real, so $4y + 12 = 0$ and $y = -3.$ The roots are then $x,$ $x + 6i,$ and $2x - 4 - 6i.$ Because the coefficients are real, the two non-real roots must be conjugates, so $2x - 4 = x,$ giving $x = 4.$ The roots are $4,$ $4 + 6i,$ and $4 - 6i.$

Now $$1 + a + b + c = P(1) = (1 - 4)\bigl(1 - (4 + 6i)\bigr)\bigl(1 - (4 - 6i)\bigr) = (-3)(9 + 36) = -135,$$ so $a + b + c = -136$ and $|a + b + c| = 136.$