Five local minima are possible. There is a quartic which has local minima in $(0, 0), (0, 1), (1, 0), (-1, -1), (1, -2)$. This example came out from a numerical search followed by a rational approximation in order to obtain a proven result (thus eliminating possible rounding issues). The following Sage code verifies the example algebraically:
R.<x, y> = QQ[]
f = (194*x^4 - 442*x^3*y + 111*x^2*y^2 + 170*x*y^3 + 29*y^4 -
364*x^3 + 576*x^2*y - 36*x*y^2 - 58*y^3 + 158*x^2 - 134*x*y
+ 29*y^2)
fx, fy = f.derivative(x), f.derivative(y)
critical_points = ideal([fx, fy]).variety()
hesse = matrix(2, 2, [fx.derivative(x), fx.derivative(y),
fy.derivative(x), fy.derivative(y)])
minima = []
for cp in critical_points:
h = hesse.substitute(cp)
if h.trace() > 0 and h.det() > 0:
minima.append(cp)
assert len(minima) == 5
Remark: Theorem 3.1.6. of the paper Critical points of real polynomials ... by Shustin seems to claim that $5$ minima are possible. However, the "proof" given in Section 3.13 merely says "Assume that $d \le 4$. Then all the index distributions except [...] can be easily realized as it was explained above".
I don't know what the author means by "above", as all the previous cases handle degrees $d\ge5$.