Five local minima are possible. There is a quartic which has local minima in $(0, 0), (0, 82), (82, 0), (-253, 593), (-649, 237)$. 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 = (46324004878805168630254223262455*x^4 +
306992632289131400429573117563212*x^3*y +
634038569228212054158919111315662*x^2*y^2 +
413369023506452188951928070902676*x*y^3 +
83898898446890528970856409397843*y^4 -
5065324255328303287915841363261976*x^3 -
25173395847708774835224995640183384*x^2*y -
33896259927529079494058101814019432*x*y^2 -
11038630209215743721318102045519592*y^3 +
69665795209396673989693247728208*x^2 +
229479129419752644122049558016717152*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 $d\ge5$.