When we complete the square, we replace all z -terms in the second-degree expression with a quadratic term which contains z, according to the formula
z2+az=z+2a2−2a2
In our case, we set a=3i in order to complete the square:
z2+3iz−41=z+23i2−23i2−41=z+23i2−49−1−41=z+23i2+2
z+2a
2−2a2 
z+23i
2−23i2−41=z+23i2−49
−1
−41=z+23i2+2