Let take a look to the special case:
$$
y=ax
$$
where $a$ is a fixed nonzero integer. When $a=1$ was already observed by Max Alexseyev that there is no solution.
Since

$$
x^4+y^4+(x+y)^4=2(x^2+xy+y^2)^2
$$
the equation $x^4+y^4+1=z^2$ can then be written:
$$
X^2 -DY^4 =1
$$
with
$$
X=z, Y=x, D = a^4+1.
$$

(Or, more simply, we get this also directly from the original equation...).

Observe now that
the equation
$$
a^4+1=w^2
$$
has no integer solution $w$.
See e.g., Corollary in page 17 of Mordell's `Diophantine Equations`

.

Then it follows from the paper of Togbe et al. below that there are at most $2$ positive solutions
$x,z$ of the equation.

Togbe, A.; Voutier, P. M.; Walsh, P. G.(3-OTTW)
Solving a family of Thue equations with an application to the equation $x^2-Dy^4=1$.
Acta Arith. 120 (2005), no. 1, 39–58.
11D59 (11D25)

Let $D$ be a positive nonsquare integer. The authors study the Diophantine equation $X^2-DY^4=1$ in positive integers $X$ and $Y$ and refine a theorem of W. Ljunggren [Skr. Norske Vid.-Akad. Oslo I 1936, no. 12, 1--73; Zbl 0016.00802]. Let $(T_1,U_1)$ be the smallest integer solution to the Pell equation $X^2-DY^2=1$. For $k\ge 1$, let $T_k+U_k\sqrt D={(T_1+U_1\sqrt D)^k}$ represent all positive integer solutions to the Pell equation. The authors prove:
There are at most two positive integer solutions $(X,Y)$ to the equation $X^2-DY^4=1$. If two solutions $Y_1 <Y_2$ exist, then $Y_1^2=U_1$ and $Y_2^2=U_2$, except only if $D=1785$ or $D=16\cdot 1785$, in which case $Y_1^2=U_1$ and $Y_2^2=U_4$. If only one positive integer solution $(X,Y)$ exists, then $Y^2=U_l$ where $U_1=lv^2$ for some squarefree integer $l$, and either $l=1$, $l=2$, or $l=p$ for some prime $p\equiv 3\pmod 4$.
The problem is reduced to solving the family of Thue equations $x^4+4tx^3y-6tx^2y^2-4t^2xy^3+t^2y^4=t_0^2$, where $t_0$ divides $t$ and $t_0\le \sqrt t$, for a positive integer $t$. However, it is not required to solve this family completely, but only for solutions whose quotient $x/y$ is near to $\beta^{(3)}$ or $\beta^{(4)}$, where $\beta^{(j)}$, $j=1,\dots,4$, denote the roots of the univariate polynomial corresponding to the Thue equation in a particular order defined in the paper. For these two roots, an effective measure of irrationality can be proved by Thue's hypergeometric method.
Reviewed by Clemens Heuberger

rationalsolutions. Elkies (Math. Comp. vol. 51, no. 184 (1988), pp. 825–835) showed that the equation $A^4+B^4+C^4=D^4$ has infinitely many integer solutions; by a well-known theorem we can't have $C=0$, so then $(A/C,B/C,D^2/C^2) gives a rational solution to the equation above. $\endgroup$14more comments