Five local minima are possible. There is a quartic which has local minima in $(0, 0), (0, -1), (-1, 0), (2, 5), (5, 2)$. 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 = (x^4 - 7*x^3*y + 16*x^2*y^2 - 7*x*y^3 + y^4 + 2*x^3 -
8*x^2*y - 8*x*y^2 + 2*y^3 + x^2 - x*y + 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)
print(len(minima) == 5)
An additional nice feature of this example is its symmetry $f(x, y)=f(y, x)$.
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 $d\ge5$.