= Solution
Choose $0<R<R_1$, $0<S<S_1$ and $0<T<T_1$. On this smaller closed <polydisc>, the <holomorphic function> $A$ and its $w$-<derivative> are bounded; write $|A|\le M$ and $|\partial_w A|\le L$. Choose $0<r<R$ with $rM<S/2$ and $rL<1$, replacing a zero bound by any positive upper bound if necessary, and choose $0<\varepsilon<T$.
Consider functions $u(z,\alpha)$ continuous on $\overline{D(0,r)}\times\overline{D(0,\varepsilon)}$, jointly <holomorphic> in the interior, with $u(0,\alpha)=0$ and $\|u\|_\infty\le S/2$. This is a closed subset of a <Banach space> in the supremum <norm>. The integral operator
$$
(\mathcal Tu)(z,\alpha)=\int_0^z A(\zeta,u(\zeta,\alpha),\alpha)\,d\zeta
$$
is well defined on the straight segment and is jointly <holomorphic>. Indeed, it equals $z\int_0^1 A(tz,u(tz,\alpha),\alpha)\,dt$, an integral of jointly <holomorphic functions> locally uniformly bounded in both variables. Convexity of the $w$-disc and the <derivative> bound give
$$
\|\mathcal Tu\|_\infty\le rM<S/2,\qquad
\|\mathcal Tu-\mathcal Tv\|_\infty\le rL\|u-v\|_\infty.
$$
Thus the <Banach fixed-point theorem> gives a fixed point $f(z,\alpha)$. More explicitly, the successive integral iterates starting from zero converge uniformly, and <locally uniform convergence of holomorphic functions> makes their limit jointly <holomorphic>. Differentiating the integral equation gives the required <complex ordinary differential equation> and initial value. This proves \b[local existence, uniqueness and joint holomorphic dependence on $(z,\alpha)$]. Uniqueness among arbitrary local solutions follows first on a small disc where their values stay within $|w|<S$, by the same <contraction mapping> estimate, and then throughout their common connected domain by the <identity theorem>.
For <holomorphic dependence of ordinary differential equations on parameters> in the initial value, fix an admissible initial value $w_*$ and put $u=f-w_0$. The equation becomes
$$
u'=A(z,u+w_0),\qquad u(0)=0.
$$
Its right-hand side is jointly <holomorphic> in $(z,u,w_0)$ near $(0,0,w_*)$. The preceding construction supplies a common disc and a jointly <holomorphic> solution for $w_0$ near $w_*$. Consequently $f=u+w_0$ is jointly <holomorphic> in $z$ and its initial value $w_0$.
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