A three-complex-parameter class of orthogonal Laurent polynomials on the unit circle associated with basic hypergeometric or
q
q
-hypergeometric functions is considered. To be precise, we consider the orthogonality properties of the sequence of polynomials
{
2
Φ
1
(
q
−
n
,
q
b
+
1
;
q
−
c
+
b
−
n
;
q
,
q
−
c
+
d
−
1
z
)
}
n
=
0
∞
\{\,_2\Phi _1(q^{-n},q^{b+1};q^{-c+b-n}; q, q^{-c+d-1}z)\}_{n=0}^{\infty }
, where
0
>
q
>
1
0 > q > 1
and the complex parameters
b
b
,
c
c
and
d
d
are such that
b
≠
−
1
,
−
2
,
…
b \neq -1, -2, \ldots
,
c
−
b
+
1
≠
−
1
,
−
2
,
…
c-b+1 \neq -1, -2, \ldots
,
R
e
(
d
)
>
0
\mathcal {R}e(d) > 0
and
R
e
(
c
−
d
+
2
)
>
0
\mathcal {R}e(c-d+2) > 0
. Explicit expressions for the recurrence coefficients, moments, orthogonality and also asymptotic properties are given. By a special choice of the parameters, results regarding a class of Szegő polynomials are also derived.