In the course of some physical considerations I came across a complex vector field

$$ mathbf u = mathbf v + i mathbf w, $$ with

begin{align}

mathbf v:& mathbb R^3to mathbb R^3\

mathbf w:& mathbb R^3to mathbb R^3

end{align}

and the special propert, that it has a divergence-free imaginary and

curl free real components, that means

begin{align}

vec{nabla}times mathbf v & = mathbf 0 tag{1}\

vec{nabla}cdotmathbf w & = 0 tag{2}

end{align}

In my attempts to better understand and interpret the quantitiy described by this field, I started to wonder if:

Question 1: This property has a special name/term in complex vector analysis.

Question 2: If there are any important prominent examples of such fields.

Question 3: If there are other properties that follow as a consequence of (1) and (2).

Question 4: If anyone can hint me at literature where such fields are investigated.

I would be very grateful for any hints at this. I would hope for answers like those to this question.