At what point the function $f(z)=|z|^2+i\bar z+1$ is differentiable?

Question

The function $f(z)=|z|^2+i\bar z+1$ is differentiable at

1. $i$
2. $1$
3. $-i$
4. no point in $\mathbb{C}$

So we need to use the definition $f'(z)=\lim_{\Delta z\to 0}\frac{f(z+\Delta z)-f(z)}{\Delta z}$, right? I can write it as $f(z)=z\bar z+i\bar z+1$. How should I proceed next? Any hint ? Thanks.

Answer 1

Hint $$f(x+iy)=(x^2+y^2+y+1)+ix$$ and use Cauchy-Riemann.