Summary: The complexification of the standard family of circle maps \({\mathbf F}_{\alpha \beta} (\theta)= \theta+\alpha+ \beta\sin (\theta) \bmod (2\pi)\) is given by \(F_{\alpha \beta} (\omega)= \omega e^{i\alpha} e^{(\beta/2) (\omega-1/ \omega)}\) and its lift \(f_{\alpha \beta} (z)=z +\alpha+ \beta\sin (z)\). We investigate the three-dimensional parameter space for \(F_{\alpha \beta}\) that results from considering \(\alpha\) complex and \(\beta\) real. In particular, we study the two-dimensional cross-sections \(\beta=\) constant as \(\beta\) tends to zero. As the functions tend to the rigid rotation \(F_{\alpha,0}\), their dynamics tend to the dynamics of the family \(G_\lambda(z) =\lambda ze^z\) where \(\lambda =e^{-i \alpha}\). This new family exhibits behavior typical of the exponential family together with characteristic features of quadratic polynomials. For example, we show that the \(\lambda\)-plane contains infinitely many curves for which the Julia set of the corresponding maps is the whole plane. We also prove the existence of infinitely many sets of \(\lambda\) values homeomorphic to the Mandelbrot set.