The aim of this paper is to study the performance of the amplitude-based model $\widehat{x} \in {\rm argmin}_{x\in \mathbb{C}^d}\sum_{j=1}^m\left(|\langle a_j,x\rangle |-b_j\right)^2$, where $b_j=|\langle a_j,x_0\rangle|+\eta_j$ and $x_0\in {\mathbb C}^d$ is a target signal. The model is raised in phase retrieval and one has developed many efficient algorithms to solve it. However, there are very few results about the estimation performance in complex case. We show that $\min_{\theta\in[0,2\pi)}\|\widehat{x}-\exp(i\theta)\cdot x_0\|_2 \lesssim \frac{\|\eta\|_2}{\sqrt{m}}$ holds with high probability provided the measurement vectors $a_j\in {\mathbb C}^d,$ $j=1,\ldots,m,$ are complex Gaussian random vectors and $m\gtrsim d$. Here $\eta=(\eta_1,\ldots,\eta_m)\in \mathbb{R}^m$ is the noise vector without any assumption on the distribution. Furthermore, we prove that the reconstruction error is sharp. For the case where the target signal $x_0\in \mathbb{C}^{d}$ is sparse, we establish a similar result for the nonlinear constrained LASSO. This paper presents the first theoretical guarantee on quadratic models for complex phase retrieval. To accomplish this, we leverage a strong version of restricted isometry property for low-rank matrices.