Five local minima are possible. There is a quartic which has local minima in $(0, 0), (0, 2), (2, 0), (6, 3), (-15, -5)$. 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 = (3189907632*x^4 - 10202120142*x^3*y + 18951229653*x^2*y^2 -
145880135028*x*y^3 + 291802357260*y^4 - 8510088512*x^3 +
20404240284*x^2*y + 291760270056*x*y^2 - 778143287520*y^3 +
11004480*x^2 + 11004480*y^2)
fx = f.derivative(x)
fy = f.derivative(y)
critical_points = ideal([fx, fy]).variety()
assert len(critical_points) == 5
hesse = matrix(2, 2, [fx.derivative(x), fx.derivative(y),
fy.derivative(x), fy.derivative(y)])
for cp in critical_points:
h = hesse.substitute(cp)
assert h.trace() > 0
assert h.det() > 0
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 $d\ge5$.