  We propose to use the complex-range Gaussian basis functions, {r^l e^{-(1 \pm
i\omega)(r/r_n)^2}Y_{lm}(\hat{r}); r_n in a geometric progression}, in the
calculation of three-body resonances with the complex-scaling method (CSM) in
which use is often made of the real-range Gaussian basis functions, {r^l
e^{-(r/r_n)^2}Y_{lm}(\hat{r})}, that are suitable for describing the
short-distance structure and the asymptotic decaying behavior of few-body
systems. The former basis set is more powerful than the latter when describing
the resonant and nonresonant continuum states with highly oscillating amplitude
at large scaling angles \theta. We applied the new basis functions to the CSM
calculation of the 3\alpha resonances with J=0^+, 2^+ and 4^+ in 12C. The
eigenvalue distribution of the complex scaled Hamiltonian becomes more precise
and the maximum scaling angle becomes drastically larger (\theta_{max}=16 deg.
\arrow 36 deg.) than those given by the use of the real-range Gaussians. Owing
to these advantages, we were able to confirm the prediction by Kurokawa and
Kato [Phys. Rev. C 71, 021301 (2005)] on the appearance of the new broad 0^+_3
state; we show it as an explicit resonance pole isolated from the 3$\alpha$
continuum.
