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{{see also|Exponentiation #Powers of complex numbers}}

The [[complex numbers|complex]] square function&nbsp;{{math|''z''²}} is a twofold cover of the [[complex plane]], such that each non-zero complex number has exactly two square roots. This map is related to [[parabolic coordinates]].<!-- unfortunately, incompatible coefficients and orientation conventions hinder a simple explanation such as σ+iτ → (σ,τ)-parabolic -->

{{anchor|{{!}}z{{!}}²}}Another, more well known, function is the square of the [[absolute value]] {{math|1=&#124; ''z'' &#124;² = ''z'' [[complex conjugate|''{{overline|z}}'']]|class=nounderlines}}, which is real-valued. It is very important for [[quantum mechanics]]: see [[probability amplitude]] and [[Born rule]]. Complex numbers form one of [[Hurwitz's theorem (composition algebras)|four possible Euclidean Hurwitz algebras]] that are defined with a real quadratic form {{mvar|q}}; here {{math|1=''q''(''z'') = &#124; ''z'' &#124;²}}. In a Euclidean Hurwitz algebra this {{mvar|q}} equals to the square of the distance to 0 discussed [[#r²|above]], and the absolute value {{math|&#124; ''z'' &#124;}} can be defined as the (arithmetical) square root of {{math|''q''(''z'')}}. Multiplicativity of {{mvar|q}} in these algebras explains (or relies upon) certain algebraic identities (see [[#Related identities|below]]).