For real-valued functions \(q,w\in L^1[-1,1]\) with \(w(x)\neq0\) for almost all \(x \in [-1,1]\), the indefinite spectral problem \[ -y''+qy=\lambda w y,\;y(-1)=y(1)=0 \tag{1} \] in \(L^2_{|w|}[-1,1]\) may have non-real eigenvalues. It is well known that if the self-adjoint problem (2) obtained from (1) by replacing \(w\) with \(|w|\) has \(n\) negative eigenvalues, then (1) has at most \(2n\) non-real eigenvalues. But in general, there is no relation between the magnitude of the negative eigenvalues of (2) and the non-real eigenvalues of (1). In this paper the authors prove bounds for the real and imaginary parts of non-real eigenvalues of (1) in terms of \(q\) and \(w\) under the additional assumption that \(xw(x)>0\) for almost all \(x \in [-1,1]\). It it also shown that if the problem (3), obtained by replacing \(w\) in (1) with \(1\), has exactly one negative eigenvalue, then (1) has two pure imaginary eigenvalues.