  A new variant of the $(2+1)$-dimensional [$(2+1)d$] Boussinesq equation was
recently introduced by J. Y. Zhu, arxiv:1704.02779v2, 2017; see eq. (3). First,
we derive in this paper the one-soliton solutions of both bright and dark types
for the extended $(2+1)d$ Boussinesq equation by using the traveling wave
method. Second, $N$-soliton, breather, and rational solutions are obtained by
using the Hirota bilinear method and the long wave limit. Nonsingular rational
solutions of two types were obtained analytically, namely: (i) rogue-wave
solutions having the form of W-shaped lines waves and (ii) lump-type solutions.
Two generic types of semi-rational solutions were also put forward. The
obtained semi-rational solutions are as follows: (iii) a hybrid of a
first-order lump and a bright one-soliton solution and (iv) a hybrid of a
first-order lump and a first-order breather.
