In Matlab, I mostly use the solver ode45 and it seems that I cannot use complex numbers when specifying the interval of integration. An oversimple example would be if I wanted to solve numerically the initial value problem $$\frac{{\mbox{d}}y}{{\mbox{d}}t}=y,\;\;y(0)=3,$$ in the complex plane on the straight line between $t=0$ and $t=3+2i$. There does not seem to be a way to use an ODE solver in MatLab that would let me specify an interval in the complex plane.
My question is: is there a way to use a Matlab ODE solver to solve an ODE in the complex plane?
To solve a ODE along a curve in the complex plane, you have to first transform it into an ODE on a real interval. Let me demonstrate this procedure using your example. We first parametrize the line from $0$ to $3 + 2 \mathrm{i}$ by $$ t = s \cdot (3 + 2 \mathrm{i}) \quad \text{for} \quad s \in [0, 1] $$ and introduce the function $$ w(s) = y(t) = y(s \cdot (3 + 2 \mathrm{i})) \,. $$ The chain rule gives the relation that $$ \frac{dw}{ds} = \frac{dy}{dt} \frac{dt}{ds} = \frac{dy}{dt} (3 + 2\mathrm{i}) \,. $$ Hence, $$ \frac{dy}{dt} = \frac{1}{3 + 2\mathrm{i}} \frac{dw}{ds} = \frac{3 - 2\mathrm{i}}{13} \frac{dw}{ds} $$ Replacing $y(t)$ and $\frac{dy}{dt}$ by $w(s)$ and $\frac{3 - 2\mathrm{i}}{13} \frac{dw}{ds}$ in the ODE, we see that $$ \frac{3 - 2\mathrm{i}}{13} \frac{dw}{ds} = w \,, \quad w(0) = 0 \,. $$ This equation is now an ODE in a real variable. Hence, you can use Matlab to solve it. Since $$ s = \frac{t}{3 + 2\mathrm{i}} \,, $$ you obtain your solution $y(t)$ by $$ y(t) = w(s) = w(\frac{t}{3 + 2\mathrm{i}}) $$ for points $t$ on the line.