I would like to calculate the left and the right Haar measures of the group \begin{equation*} G = \left \{ \begin{pmatrix} a & b \\ & 1 \end{pmatrix} \:| \: a \in \mathbb{C}^\times, b \in \mathbb{C} \right \} \end{equation*} and its modular character.
I have seen this calculation for the affine group in one dimension given by \begin{equation*} H = \left \{ \begin{pmatrix} a & b \\ & 1 \end{pmatrix} \:| \: a > 0, b \in \mathbb{R} \right \}, \end{equation*} however I am stuck in this complex setting. I was hoping that you could explain how to carry out this calculation.
Thank you very much in advance.
Suppose there is some density on $G=\mathrm{Aff}_1\mathbb{C}$ which integrates to the measure:
$$ \mu(X)=\int_X \rho(x)\,\mathrm{d}x . $$
(Here, $\mathrm{d}x$ is from the usual Lebesgue measure on $\mathbb{C}^{\times}\times\mathbb{C}$ interpreted as a subspace of $\mathbb{R}^4$.)
Then left-invariance $\mu(X)=\mu(gX)$ may be expressed as
$$ \begin{array}{ll} \displaystyle \int_X \rho(x)\,\mathrm{d}x & \displaystyle =\int_{gX} \rho(x)\,\mathrm{d}x \\[5pt] & \displaystyle =\int_X \rho(gx)\,\mathrm{d}(gx) \\[5pt] & \displaystyle = \int_X\rho(gx)J_{g,x}\,\mathrm{d}x \end{array} $$
where we use a change of variables and $J_{g,x}$ is the Jacobian determinant of left-multiplication-by-$g$. Since this is true for all $X$, the integrands must match, so set $x=e$ and $\rho(e)=1$ so $\rho(g)=J_{g,x}^{-1}$.
Set $g=(a,b)=(a_1+a_2i,b_1+b_2i)$ and $x=(x_1+x_2i,y_1+y_2i)$ so that
$$ gx=\big((a_1x_1-a_2x_2)+(a_2x_1+a_1x_2)i,(a_1y_1-a_2y_2+b_1)+(a_2y_1+a_1y_2+b_2)i\big) $$
with corresponding determinant
$$ J=\det\begin{pmatrix} a_1 & -a_2 & 0 & 0 \\ a_2 & a_1 & 0 & 0 \\ 0 & 0 & a_1 & -a_2 \\ 0 & 0 & a_2 & a_1 \end{pmatrix}=(a_1^2+a_2^2)^2. $$
So we conclude $\rho(a,b)=1/|a|^4$.
I'll let you figure out the right-invariant Haar measure and modular function.